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% Obscured Symmetry Breaking
% and LowLying Excited States (Full Paper)
% by Tohru Koma and Hal Tasaki
% Department of Physics, Gakushin University,
% Mejiro, Toshimaku, Tokyo 171, Japan
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\begin{center}
{\Huge\bf
Obscured Symmetry Breaking\\
and\\
LowLying States\\
in\\
Quantum ManyBody Systems\\
}
\bigskip\bigskip
{\large
Tohru Koma\\
\bigskip
Hal Tasaki\\}
\bigskip
{\it Department of Physics\\
Gakushuin University\\
Mejiro, Toshimaku, Tokyo 171\\
JAPAN}
\end{center}
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\vfil\vfil
\noindent
We consider a quantum manybody system
on a lattice which exhibits a
spontaneous symmetry breaking in its infinite volume ground states,
but in which
the corresponding order operator does not commute with the
Hamiltonian.
In the corresponding finite system, the symmetry breaking is usually
``obscured'' by ``quantum fluctuation''
and one gets a symmetric ground
state with a long range order.
In such a situation, Horsch and von der Linden proved
that the finite system has a lowlying
eigenstate
whose energy per site converges to the ground state energy per site
as the system size increases.
When the system has a continuous symmetry, we prove that the
number of independent lowlying eigenstates grows faster than
any given small
order
of the system size.
We show that a translation invariant lowlying state converges
to a ground state in the infinite volume limit.
We also construct infinite
volume ground states with explicit symmetry breaking by taking
linear combinations of the (finitevolume) ground state and
the lowlying states, and then taking infinite volume limits.
We conjecture these infinite volume ground states
to be pure.
Our general theorems do not only shed light on the nature of
symmetry breaking in quantum manybody systems, but
provide indispensable information for numerical approaches
to these systems.
We also discuss applications of our general results to a
variety of interesting examples.
\par\noindent
\bigskip
\hrule
\bigskip
\noindent
{\bf KEY WORDS:} Symmetry breaking;
long range order;
finite systems;
quantum fluctuation;
ground states;
lowlying states;
pure states.
\vfil}\newpage
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\tableofcontents
\newpage
\Section{Introduction}
\subsection{Motivations}
Symmetry breaking in quantum manybody systems is a challenging
problem in theoretical physics.
In some situations, strong ``quantum effects'' lead to phenomena
which are hard to predict or understand from ``classical'' points of
view.
The present paper is devoted to one of such ``quantum effects'',
namely, how a symmetry breaking manifests itself in a {\em finite\/}
system when the order operator and the Hamiltonian do not
commute with each other.
The topic reflects subtlety of both quantum mechanics and
manybody
problems, and indeed has been a source of some confusions in the
field\footnote{
See footnotes in Section~1.2 for what the confusions are.
}.
We have tried in the present paper not only to present our new
theorems, but
also to review (hopefully in an accessible manner) some background
materials
necessary to understand the nature of the problems.
Suppose that we have a quantum manybody system which exhibits
a spontaneous symmetry breaking in its infinite volume ground
states.
When the operator that measures the symmetry does not commute
with the Hamiltonian, one encounters strong ``quantum
fluctuation''.
In the corresponding {\em finite\/} system, the symmetry breaking
is usually ``obscured'' by the fluctuation, and one only gets a unique
ground state with perfect symmetry.
The ``obscured symmetry breaking'' manifests itself in the following
two different ways.
\begin{itemize}
\item
One observes a long range order in the ground state twopoint
correlation function for the order operators.
\item
There appear lowlying eigenstates whose energies per
site converge to the ground state energy per site as the system
size increases.
\end{itemize}
Although one might be tempted to interpret these lowlying
eigenstates
as counterparts of excited states in the infinite volume system,
they
are actually ``parts'' of the infinite volume ground states.
In the infinite volume limit, the lowlying
eigenstates and the unique
ground state are linearly combined to form a set of {\em pure\/}
ground states with explicit symmetry breaking.
When a continuous symmetry is broken, there are infinitely many
{\em pure\/} ground states in the infinite volume.
It is then expected that the number of independent
lowlying eigenstates in the
corresponding finite system increases indefinitely as the system
size gets larger.
The existence of such ever increasing numbers of lowlying states
has been discussed mainly by practitioners of numerical exact
diagonalization of quantum spin systems.
(It is not easy to list up all the relevant references.
See, for example,
\cite{KaplanHorschLinden,Kikuchi,Bernu,Azaria,Leung} and the references
therein.)
But rigorous information has been lacking
except in the meanfield model
\cite{LiebMattis,Kaiser,KaplanLinden}.
See also \cite{Anderson} for early related discussion within the
framework of the spin wave approximation.
The purpose of the present paper is to state general
theorems for lattice quantum manybody systems
which clarify the relation between the above mentioned
two types of manifestations of an ``obscured symmetry breaking''.
In short, our theorems show that, whenever one has a finite volume
ground state which does not break symmetry but whose correlation
function exhibits a certain long range order, one must also have
lowlying eigenstates.
The simplest of such results (see Theorem~\ref{M=1theorem}
below)
was proved by
Horsch and von der Linden \cite{HorschLinden}.
In general there is a lowlying eigenstate whose excitation energy
per
site is less than of order $N^{2}$, where $N$ is the number of
sites
in the lattice.
Our new results deal with the cases where
the long range order is related to a continuous $U(1)$
symmetry.
We show that the number of independent lowlying
eigenstates
increases faster than any given small order of $N$
as the system size $N$ increases.
This is the first rigorous (and explicit) demonstration that
ever increasing numbers of lowlying states indeed exist.
We show that any translation invariant lowlying state
converges to a ground state in the infinite volume limit.
We also construct infinite volume ground states with explicit
symmetry breaking by taking suitable linear combinations of the
lowlying states, and then taking infinite volume limits.
We conjecture that these ground states are pure, {\em i.e.\/}, are
physically natural ground states.
We also discuss applications of our general results to
some concrete examples of interest.
We believe that these results do not only clarify the nature of
symmetry breaking phenomena in quantum systems,
but also provide indispensable information for numerical
approaches to various quantum manybody systems.
The present paper is organized as follows.
In the following Section~1.2, we illustrate some of the basic
notions by studying a concrete example of the Ising model under
a transverse magnetic field.
We have tried to make this section accessible
to the readers who are not
familiar with mathematical approaches to quantum manybody
problems.
In Section~2, we state our theorems in the most general setting,
and discuss their physical consequences.
In Section~3, we discuss applications of our theorems
to typical problems.
The examples include Heisenberg
antiferromagnet, the BoseEinstein condensation in hard core Bose
gas on a lattice,
the superconductivity in lattice electron
models, and the Haldane gap problem in $S=1$ quantum antiferromagnetic
chain.
An interesting observation in the application to the Bose gas is
that, by following our general discussion in Section~2, we are naturally led
to consider ground states which do not conserve the particle number.
Sections~4 and 5 are devoted to the proofs of our theorems.
In the three Appendices, we prove and summarize some useful
results closely related to the main body of the paper.
In Appendix~\ref{APgs}, we discuss relations between three different
definitions of infinite volume ground states, and show that
they are all equivalent when restricted to translation invariant
states.
In Appendix~\ref{APpure}, we concentrate on a system with
spontaneously broken discrete symmetry, and present a theorem
which shows how to construct a pure infinite volume ground
state.
In Appendix~\ref{APfluc}, we prove lemmas which characterize
the fluctuation of bulk quantities.
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\subsection{Obscured symmetry breaking and lowlying states
in a simple example}
Before discussing general theorems, we want to make clear what
we mean by ``obscured symmetry breaking'' and ``lowlying
states'', and how these notions are related to phenomena of
symmetry breaking.
For this purpose, we shall
discuss one of the simplest models
in which one observes an ``obscured symmetry breaking'' and
``lowlying states''.
In the course, we briefly review the notions of ground states in an
infinite system, of pure states, and of symmetry breaking in
the absence of a symmetry breaking field.
Although such materials form standard background in
mathematical approaches to quantum manybody systems, we have
noted that they are not widely appreciated in standard physics
literature.
Here we will try to explain basic physical ideas rather than
developing
precise mathematical formalism.
Mathematical details will be supplied in the following sections.
Consider the $d$dimensional $L\times\cdots\times L$ hypercubic
lattice $\Lambda \subset {\bf Z}^d$, and impose periodic boundary
conditions.
We define the $S=1/2$
spin system on $\Lambda$ with the Hamiltonian
\begin{equation}
\ham = \sum_{\langle x,y \rangle} S^{(3)}_{x}S^{(3)}_{y} 
B\sum_xS^{(1)}_{x},
\label{Ising1}
\end{equation}
where the first sum is over nearest neighbor pairs of sites in
$\Lambda$, the magnetic
field satisfies $B\ge0$, and
${\bf S}_x=(S^{(1)}_x,S^{(2)}_x,S^{(3)}_x)$ denote the $S=1/2$ spin
operators at site $x$.
The model is known as the Ising model under transverse magnetic
field.
The ground state of the Hamiltonian (\ref{Ising1}) is known to
exhibit a phase transition as the transverse field $B$ is varied.
This is most clearly seen from the following behavior of the order
parameter $m(B)$.
Let $\PL(B,B')$ be the normalized ground state of the Hamiltonian
$\hamB'\op$, where $\op$ is the order operator
\begin{equation}
\op=\sum_{x\in\Lambda}S^{(3)}_x.
\label{opIsing}
\end{equation}
Define the order parameter by
\begin{equation}
m(B):=\lim_{B'\downarrow0}\lim_\TDL\frac{1}{N}
(\PL(B,B'),\op\,\PL(B,B')),
\end{equation}
where $N=L^d$ is the number of sites in $\Lambda$.
Throughout the present paper, the symbol $:=$ signifies definition.
It can be proved that, for a fixed dimension $d$, the order operator
satisfies $m(B)=0$ for sufficiently large $B$, and $m(B)>0$ for
sufficiently small $B$.
In the latter case, the global updown symmetry of the system
is spontaneously
broken in the infinite volume ground state.
Let us see how this symmetry breaking manifests itself in {\em
finite} systems.
When $B=0$, the model is nothing but the classical Ising model.
The Hamiltonian (\ref{Ising1}) has two ground states
$\PL^+$ and $\PL^$, in which all the spins are pointing up and
down,
respectively.
The ground states are ordered, and breaks the updown
symmetry of the Hamiltonian.
When $B>0$, we encounter ``quantum fluctuation''.
An elementary application of the
PerronFrobenius theorem (as in \cite{LiebMattis})
implies that the ground state $\PL(B)$
of the
Hamiltonian (\ref{Ising1}) is unique for an arbitrary finite $L$.
Hence the global updown symmetry remains unbroken in the finite
volume ground state $\PL(B)$ for any value of $B>0$.
When $m(B)>0$, we might say
that the symmetry breaking in the infinite volume limit is
``obscured'' by ``quantum fluctuation'' in finite systems.
A sign of the ``obscured symmetry breaking'' can be found
as a long range order in the
twopoint correlation functions.
Although we have $(\PL(B),S^{(3)}_x\,\PL(B))=0$ for any $B>0$,
we expect (and can prove for sufficiently small $B$) that
\begin{equation}
(\PL(B),S^{(3)}_xS^{(3)}_y\,\PL(B)) \simeq m(B)^2
\label{LROIsing}
\end{equation}
holds for sufficiently large $xy$.
Another sign of the ``obscured symmetry breaking'' can be found if
we consider the first excited state $\PL^{(1)}(B)$ of $\ham$ and
its energy $E^{(1)}_\Lambda$.
When we have $m(B)>0$, we expect
\begin{equation}
\frac{1}{N}\rbk{\EL^{(1)}\EL^{(0)}}\approx \exp[\tau(B) L^d]
\label{expdecay}
\end{equation}
holds as $L\toinf$ with a positive finite constant $\tau(B)$,
where $\EL^{(0)}$ denotes the ground state energy.
The sequence of states $\{\PL^{(1)}(B)\}_\Lambda$ in this situation
are typical examples of lowlying eigenstates\footnote{
A beginner of exact diagonalization
might identify $\PL^{(1)}(B)$ as a finitesize counterpart of
an excited state in the infinite system.
As will become clear soon, this is totally misleading.
}.
The ground state $\PL(B)$ and the first excited state
$\PL^{(1)}(B)$
inherit the existence of symmetry breaking in the
infinite volume limit.
We expect that, when $m(B)>0$, these states can be written as
\begin{equation}
\PL(B)\simeq\frac{1}{\sqrt{2}}
\rbk{\tilde{\Phi}^+_\Lambda(B)+\tilde{\Phi}^_\Lambda(B)},
\label{PL=}
\end{equation}
and
\begin{equation}
\PL^{(1)}(B)\simeq\frac{1}{\sqrt{2}}
\rbk{\tilde{\Phi}^+_\Lambda(B)\tilde{\Phi}^_\Lambda(B)},
\label{PL1=}
\end{equation}
where $\tilde{\Phi}^+_\Lambda$ and $\tilde{\Phi}^_\Lambda$ are
the states obtained by taking into account local quantum
fluctuations into the completely ordered states
$\PL^+$ and $\PL^$, respectively.
We can now discuss how the lowlying eigenstates are related to
the symmetry breaking in the infinite volume limit.
A ground state in the infinite system may be defined by the
thermodynamic limit
\begin{equation}
\omega_B(A):=\lim_\TDL(\PL(B),A\,\PL(B)),
\label{omegaB}
\end{equation}
where $A$ is an arbitrary local operator ({\em i.e.}, a polynomial of
spin operators), and $\PL(B)$ is the unique ground state of $\ham$.
The limit is welldefined if one takes suitable subsequence of
lattices.
(See Section~2.5 and Appendix~\ref{APgs} for details.)
We shall define the ground state energy density as
$\epsilon_0:=\omega_B(h_x)$.
Here the local Hamiltonian is
$h_x=\sum_{y;xy=1}S^{(3)}_xS^{(3)}_y/2BS^{(1)}_x$,
where the sum runs over the sites $y$ which are neighboring to $x$.
Since the finite volume ground state $\PL(B)$ respects the global
updown symmetry, so does the infinite volume ground state
$\omega_B(\cdots)$.
In particular we have
\begin{equation}
\omega_B(S^{(3)}_x)=0,
\label{NOSB0}
\end{equation}
for any $x$.
One might suspect from the above construction
and the relation (\ref{NOSB0}) that, when the
symmetry breaking field $B'$ is vanishing,
there is no
symmetry breaking even in the infinite volume limit\footnote{
This is another possible confusion,
to which we sometimes encounter.
}.
>From a physical point of view, however, this conclusion is
unnatural and misleading.
One should recall that
there are many situations in nature where we do
observe a symmetry breaking in the absence of explicit symmetry
breaking fields\footnote{
A typical example is antiferromagnetism, in which a staggered
magnetic field plays the role of a symmetry breaking field.
No mechanism can generate a real staggered magnetic field in an
antiferromagnetic material!
A more drastic example is the BoseEinstein condensation, where
the symmetry breaking field should create and annihilate particles!!
}.
It is indeed possible to develop mathematically sensible definitions
of infinite volume ground states which are capable of describing a
symmetry breaking without symmetry breaking
fields.
We discuss precise definitions in Section~2.5
(Definition~\ref{DEFgs}) and
Appendix~\ref{APgs}.
Here we shall see concrete examples.
Before discussing the symmetry breaking, however,
let us observe that the above
ground state $\omega_B(\cdots)$ indeed has an unnatural property.
Let $\Omega$ be a hypercubic region in ${\bf Z}^d$, and denote by
$\Omega$ the number of sites in $\Omega$.
Consider the bulk physical quantity
$M_\Omega :=\sum_{x\in\Omega}S^{(3)}_x$.
By combining (\ref{LROIsing}), (\ref{NOSB0}),
and Lemma~\ref{flucLemma1}, we find that
\begin{equation}
\frac{1}{\Omega^2}
\omega_B\sbk{\rbk{M_\Omega\omega_B(M_\Omega)}^2}
\ge
m(B)^2,
\label{nocluster}
\label{cof}
\end{equation}
as $\Omega\toinf$.
The relation (\ref{nocluster}) implies that, in the state $\omega_B(\cdots)$,
the intensive bulk quantity $M_\Omega/\Omega$ has a finite fluctuation.
This is in contrast to the basic requirement in physics that any intensive
bulk
quantity exhibits essentially no fluctuation in a thermodynamically stable
phase.
An infinite volume state in which any
intensive bulk quantity has vanishing fluctuation
is called a {\em pure} state\footnote{
The notion of {\em pure} states in an infinite system is distinct
from that in ordinary quantum mechanics.
See Definition~\ref{DEFpure} and the remark following it.
}.
It is believed that a physically realizable state in a large system
can be well approximated by a pure state.
The behavior (\ref{nocluster}) implies that the state
$\omega_B(\cdots)$
is not pure, and is hence unphysical.
Then there must be some physically natural ground states.
Equations (\ref{PL=}) and (\ref{PL1=}) motivate us to
define two states in the infinite system by
\begin{eqnarray}
\omega^\pm_B(A)&:=&\lim_\TDL
\frac{1}{2}\rbk{(\PL(B)\pm\PL^{(1)}(B)),A
\,(\PL(B)\pm\PL^{(1)}(B))},
\ret
&\simeq&\lim_\TDL
(\tilde{\Phi}^\pm_\Lambda(B),A\,\tilde{\Phi}^\pm_\Lambda(B)),
\label{omegapm}
\end{eqnarray}
with an arbitrary local operator $A$.
By using (\ref{expdecay}), the translation invariance of the expectation
values, and the fact that
$(\PL^{(1)}(B),\ham\PL(B))=0$,
we find that these states satisfy
\begin{equation}
\omega^\pm_B(h_x)=\epsilon_0,
\label{GS0}
\end{equation}
for any neighboring $x$ and $y$,
where $\epsilon_0=\omega_B(h_x)$ is the
previously defined ground state energy density.
Following Definition~\ref{DEFgs},
we shall interpret the relation (\ref{GS0}) as indicating that the
states $\omega^\pm_B(\cdots)$ are infinite volume ground
states\footnote{
The existence of ground states other than $\omega_B(\cdots)$
apparently contradicts with the ``uniqueness of the ground
state'' we mentioned earlier, and has been a source of confusion
(especially in much more delicate situations, e.g., in Heisenberg
antiferromagnets).
Of course there is no contradiction, since the uniqueness (as is
proved by the PerronFrobenius argument \cite{LiebMattis})
applies only to
a finite system.
}.
We stress that this is a natural definition of ground states.
In a bulk (or an infinite) system, it is no longer meaningful to talk
about small difference in the total energy.
What really count are the expectation values of the local
Hamiltonian, and the present definition is designed precisely to
look only at them.
We shall discuss more about the definitions of infinite volume
ground states in Appendix~\ref{APgs}.
The final expression in (\ref{omegapm}) suggests
the existence of an explicit symmetry breaking as
\begin{equation}
\omega_B^\pm(S^{(3)}_x)=\pm m(B).
\label{oMmM}
\end{equation}
If we assume the existence of a long range order as in
(\ref{LROIsing}) and the existence of a gap
above $E_\Lambda^{(1)}$, we can prove the relation (\ref{oMmM})
from (\ref{OrderAP}) and Lemma~\ref{1=PsiLemma}.
Under the same assumptions, we can also prove that
the infinite volume ground state
$\omega_B^\pm(\cdots)$ are pure.
See Theorem~\ref{pureth}.
We conclude that $\omega_B^\pm(\cdots)$
constructed by taking
linear combinations of $\PL(B)$ and $\PL^{(1)}(B)$
are the physically natural ground states in the infinite volume.
Let us summarize what we have learned from the
present simple example.
When there is an obscured symmetry breaking, we have the following.
\begin{itemize}
\item
There inevitably
exists a lowlying state.
\item
The infinite volume ground state defined by a naive infinite volume
limit of finite volume ground states is not ``pure'', i.e., is
unphysical.
\item
A pure ground state may be formed by taking a linear combination of
the finite volume ground state and the lowlying state, and then
taking the infinite volumelimit.
\end{itemize}
In Section~2, we will see that these features are typical
when there is an ``obscured symmetry breaking''.
We will mainly concentrate on
how the situation is modified when the relevant symmetry
is a continuous one.
\bigno
{\bf Remark:}
The reader might wonder about the nature of the ground state
$\PL(B)$ and the first eigenstate $\PL^{(1)}(B)$ when $B$ is large
enough to have $m(B)=0$.
In this case, we expect that $\ham$ has a finite gap almost uniform
in the lattice size $N$, and thus
\begin{equation}
\frac{1}{N}\rbk{\EL^{(1)}\EL^{(0)}}\approx \frac{1}{N},
\end{equation}
as $N\toinf$.
Although the state $\PL^{(1)}(B)$ may be again called a lowlying
eigenstate, its nature is totally different from that in the case
with
$m(B)>0$.
Roughly speaking, the first excited state $\PL^{(1)}(B)$ can be
regarded
as the state in which a single ``magnon'' is in the $k=0$ state, {\em
i.e.\/},
\begin{equation}
\PL^{(1)}(B)\simeq\sum_{x\in\Lambda}\PL^{(x)}(B),
\end{equation}
where $\PL^{(x)}(B)$ is the state in which the magnon is localized at
the
site $x$.
When $B$ is extremely large, $\PL^{(x)}(B)$ may be approximated by
the
state in which the spin at $x$ is pointing in the direction opposite to
the magnetic field and all the other spins are pointing in the
direction of the field.
The biggest difference from the case with $m(B)>0$ is that the
limit
\begin{equation}
\tilde{\omega}_B(\cdots):=\lim_\TDL
((\alpha\PL(B)+\beta\PL^{(1)}(B)),(\cdots)\,
(\alpha\PL(B)+\beta\PL^{(1)}(B))),
\end{equation}
with any $\alpha$, $\beta$ with $\alpha^2+\beta^2=1$, defines
exactly the same state as $\omega_B(\cdots)$ in (\ref{omegaB}).
More precisely, we have
\begin{equation}
\omega_B(A)=\tilde{\omega}_B(A),
\label{omega=tilomega}
\end{equation}
for an arbitrary local operator $A$.
The equality (\ref{omega=tilomega}) should be expected since, in an
infinite system with only a single magnon, the probability of
observing the magnon is vanishing.
One may form a state distinct from $\omega_B(\cdots)$ by
considering, e.g., the limit
\begin{equation}
\omega^{(x)}_B(\cdots):=\lim_\TDL
(\PL^{(x)}(B),(\cdots)\,\PL^{(x)}(B)),
\end{equation}
but the resulting state is not translation invariant.
We clearly see $\omega^{(x)}_B(h_x)>\epsilon_0$, and hence
$\omega^{(x)}_B(\cdots)$ is not an infinite volume ground state.
We expect that, in this case, the infinite volume ground state is
unique and preserves the global updown symmetry.
Such a result can be proved rigorously for sufficiently large $B$.
See Theorem~\ref{UniqueGS}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Results and physical consequences}
In the present section, we describe our main results and their
physical consequences in a general setting.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Theorem of Horsch and von der Linden}
Before discussing our own results, we describe the theorem due to
Horsch and von der Linden\cite{HorschLinden},
which is the first rigorous result
concerning the existence of a lowlying state in the presence of an
obscured symmetry breaking.
We consider a quantum system
on a finite lattice $\Lambda$ with $N$ sites.
With each site $x \in \Lambda$, we associate a finitedimensional
Hilbert space ${\cal H}_x$.
The full Hilbert space is
\begin{equation}
{\cal H}_\Lambda := \bigotimes_{x \in \Lambda} {\cal H}_x.
\label{Hilb}
\end{equation}
We note that a ``site'' in $\Lambda$ need not be an atomic site of a quantum
manybody system.
If necessary, one may call a group of atomic sites a ``site'', and
let $\calH_x$ be the corresponding finite dimensional Hilbert
space.
Throughout the present paper, the norm of a state
$\PsL \in {\cal H}_\Lambda$ is defined as
$\Vert \PsL \Vert ={(\PsL,\PsL)}^{1/2}$,
and
the norm of an operator $A$ on ${\cal H}_\Lambda$ as
\begin{equation}
\Vert A \Vert := \sup_{\PsL \in {\cal H}_\Lambda}
{\Vert A\PsL \Vert \over \Vert \PsL \Vert}.
\label{DEFnorm}
\end{equation}
For a fixed $\Lambda$, we take the Hamiltonian
\begin{equation}
\ham := \sum_{x \in \Lambda} \> h_x,
\label{ham}
\end{equation}
where each $h_x$ is a selfadjoint operator on ${\cal H}_\Lambda$.
Let
\begin{equation}
\op:=\sum_{x\in\Lambda}o_x
\label{ord}
\end{equation}
be the order operator, where each $o_x$ is a selfadjoint operator
on ${\cal H}_\Lambda$.
Assume that $\norm{h_x}\le h$ and $\norm{o_x}\le o$ hold
for any $x$
with finite constants $h$ and $o$.
Assume also that
$[o_x,o_y]=0$ holds for any $x,y$, and $[h_x,o_y]=0$
holds unless $y\in{\cal S}_x$.
The number of sites in the
support set ${\cal S}_x$ is bounded from above by an integer $r$.
Let $\PL$ be an eigenstate of $\ham$ with the eigenvalue $\EL$.
We assume that the state $\PL$ exhibits an ``obscured symmetry
breaking'' in the sense that it satisfies
\begin{equation}
\bkt{\op}=0,
\label{NOorder0}
\end{equation}
and
\begin{equation}
\bkt{(\op)^2}\ge (\mu oN)^2
\label{LRO0}
\end{equation}
with a constant $0<\mu\le1$.
We define
\begin{equation}
\PsL:=\frac{\op\PL}{\norm{\op\PL}},
\end{equation}
which is welldefined since $\norm{\op\PL}$ is
nonvanishing because of (\ref{LRO0}).
Then the theorem of Horsch and von der Linden is the following.
\begin{theorem}
The expectation value of the energy in the state $\PsL$ satisfies
\begin{equation}
\frac{1}{N}\abs{(\PsL,\ham\,\PsL)
\EL}
\le c_0\frac{1}{N^2}
\end{equation}
with $c_0=2r^2h\mu^{2}$.
When $\PL$ is a ground state of $\ham$, there exists at least one
eigenstate of $\ham$ whose energy $E^{(1)}_\Lambda$ satisfies
\begin{equation}
\frac{1}{N}\rbk{E^{(1)}_\Lambda\EL}\le c_0\frac{1}{N^2}.
\end{equation}
\label{M=1theorem}
\end{theorem}
\begin{proof}{Proof}
The first part follows by observing that
\begin{eqnarray}
\abs{(\PsL,\ham\,\PsL)
\EL}
&=&
\frac{\abs{\bkt{\sbk{\sbk{\op,\ham},\op}}}}{2\norm{\op\PL}^2}
\ret
&\le&\frac{\norm{\sbk{\sbk{\op,\ham},\op}}}{2\norm{\op\PL}^2}
\ret
&\le&
\frac{4r^2ho^2N}{2(\mu oN)^2}=c_0\frac{1}{N},
\end{eqnarray}
where we have used the assumed commutation relations
and the norm bounds of $h_x$
and $o_x$, as well as (\ref{LRO0}).
To prove the second part, we note that the relation
(\ref{NOorder0}) implies that the state $\PsL$ is orthogonal to the
ground state $\PL$.
Then the statement in the theorem is a simple consequence of the
variational principle.%
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Main theorems}
Now we shall describe our extensions of the above theorem.
Our theorems apply to a system with a continuous symmetry, and
establish the existence of ever increasing numbers of lowlying
eigenstates.
We again consider the finite lattice $\Lambda$ with $N$ sites,
the Hilbert space
${\cal H}_\Lambda$ as in (\ref{Hilb}), and the Hamiltonian $\ham$
as in (\ref{ham}).
We further require that the system possesses a global $U(1)$
symmetry,
whose generator is a selfadjoint operator $\gen$.
We assume that
\begin{equation}
\sbk{\ham,\gen}=0.
\end{equation}
We introduce the order operators
\begin{equation}
O_\Lambda^{(\alpha)}:=\sum_{x\in\Lambda}o_x^{(\alpha)},
\label{orderOp}
\end{equation}
where $\alpha=1,2$, and each $o_x^{(\alpha)}$ is a selfadjoint
operator on ${\cal H}_\Lambda$.
The order operators form two components of a vector
$(\opone,\optwo)$ which transforms under the action of $U(1)$,
and measure a possible spontaneous breakdown of
the $U(1)$ symmetry.
They satisfy the standard commutation relations
\begin{equation}
\sbk{\opone,\gen}=i\optwo,\quad
\sbk{\optwo,\gen}=i\opone.
\label{commutation1}
\end{equation}
We also introduce
\begin{equation}
O^\pm_\Lambda:=\opone\pm i\optwo,
\label{Oplusminus}
\end{equation}
which satisfy the commutation relations
\begin{equation}
\sbk{\opplus,\gen}=\opplus,\quad
\sbk{\opminus,\gen}=\opminus.
\label{commutation2}
\end{equation}
The operators $\opplus$ and $\opminus$ are the raising
and the lowering operators, respectively, for the quantum number
defined by the selfadjoint operator $\gen$.
We assume that these operators satisfy the following
three conditions.
\par\bigskip
\noindent
i) \quad
$[o_x^{(\alpha)},o_y^{(\beta)}]=0$ for $x\ne y$ and
$\alpha,\beta=1,2$.
\par\bigskip
\noindent
ii) \quad
$ [h_x, o_y^{(\alpha)}] = 0 $
holds for $\alpha=1,2$ unless $ y \in{\cal S}_x$.
The number of sites in the support set
${\cal S}_x \subset \Lambda$ is
bounded from above by an integer $r\geq2$.
\par\bigskip
\noindent
iii) \quad
There are finite constants $h$ and $o$, and we have
$\Vert h_x \Vert \le h$
and
$\Vert o_x^{(\alpha)} \Vert \leq o$
for any $x \in \Lambda$ and $\alpha=1,2$.
\par\bigskip
Let $\PL$ be a normalized simultaneous eigenstate
of the Hamiltonian $\ham$ and the selfadjoint operator
$C_\Lambda$.
We denote by $\EL$ the corresponding eigenvalue of $\ham$.
Usually we take $\PL$ as a ground state of $\ham$.
We assume that
\par\bigskip
\noindent
iv) \quad
the state $\Phi_\Lambda$ exhibits a long range order
in the sense that
\begin{equation}
\bkt{(\opone)^2}=\bkt{(\optwo)^2}
\geq(\mu o N)^2
\label{LRO}
\end{equation}
holds with a constant $0<\mu\le1$.
\par\bigskip
>From the commutation relations (\ref{commutation2}) and the fact
that $\PL$ is an eigenstate of $\gen$, we automatically have
\begin{equation}
\bkt{\opone}=\bkt{\optwo}=0.
\label{NOSB}
\end{equation}
In other words, the state $\PL$ have vanishing order parameters.
The relations (\ref{LRO}) and (\ref{NOSB}) together imply that the
state
$\PL$ exhibits an ``obscured symmetry breaking''\footnote{
In fact (\ref{LRO}) and (\ref{NOSB}) may well hold in a finite system
whose infinite volume limit does not exhibit a symmetry breaking,
in which case the parameter $\mu$ vanishes as $\TDL$.
What we really mean by an ``obscured symmetry breaking'' is
that (\ref{LRO}) and (\ref{NOSB}) are valid with a
$\Lambda$independent $\mu>0$.
}.
For a nonvanishing integer $M$, we consider the state
\begin{equation}
\PsL^{(M)}:=\frac{(\opplus)^M\PL}
{\norm{(\opplus)^M\PL}},
\label{Psi}
\end{equation}
where we set $(\opplus)^M=(\opminus)^{M}$ for a negative $M$.
Although the state (\ref{Psi}) is illdefined if
$(\opplus)^M\Phi_\Lambda =0$,
the following theorems guarantee that this is not the case when
certain conditions are met.
The first theorem of the present paper is the following.
\begin{theorem}
When the assumptions~i)iv) are valid, and we further have
\begin{equation}
N\ge \rbk{\frac{4r}{\mu}}^2,
\label{N>}
\end{equation}
and
\begin{equation}
\frac{\abs{M}}{N}\le\frac{\mu^2}{8r},
\label{M<}
\end{equation}
the state $\PsL^{(M)}$ of (\ref{Psi}) is welldefined.
The expectation value of the
energy in the state satisfies
\begin{equation}
\frac{1}{N}\left(\PsL^{(M)} , \ham
\PsL^{(M)})
 \EL\right
\le c_1\frac{M}{N},
\label{B2}
\end{equation}
where $c_1$ is a constant which depends only on $h$, $r$, and
$\mu$.
\label{generalMtheorem}
\end{theorem}
The theorem will be proved in Section~4.
In order to get a better estimate of the energy difference (at least
for small enough $M$), we require higher symmetry.
\par\bigno
v)\quad
The order operators satisfy the commutation relation
\begin{equation}
\sbk{\opone,\optwo}=i\gamma\gen,
\label{commutation3}
\end{equation}
where $\gamma$ is a real constant.
\par\bigskip
In other words, we assume that the vector
$(\gamma^{1/2}\opone,\gamma^{1/2}\optwo,\gen)$ form
generators of $SU(2)$.
We do not, however, assume that the system has a full $SU(2)$
symmetry.
We only require a partial $U(1)\times{\bf Z}_2(\cong O(2))$ symmetry as
follows.
\par\bigskip
\noindent
vi)\quad
We have
$U_\Lambda H (U_\Lambda)^{1}=H$ and $U_\Lambda\PL\propto\PL$,
where $U_\Lambda=\exp[i(\pi/\sqrt{\gamma})\opone]$ represents
the $\pi$rotation around the first axis.
\par\bigskip
Then the second theorem is as follows.
\begin{theorem}
When the assumptions i)vi) are valid, and we further have
\begin{equation}
\frac{M^2}{N}\le c_2,
\label{M/N<}
\end{equation}
with $c_2=\min\{\mu^2/(192r),o\mu/\sqrt{24\gamma}\}$,
the state $\PsL^{(M)}$ of (\ref{Psi}) is welldefined.
The expectation value of the
energy in the state satisfies
\begin{equation}
\frac{1}{N}\left(\PsL^{(M)} , \ham
\PsL^{(M)})
 \EL\right
\le c_3\rbk{\frac{M}{N}}^2,
\end{equation}
where $c_3$ is a constant which depends only on
$h$, $o$, $r$, $\mu$,
and $\gamma$.
\label{smallMtheorem}
\end{theorem}
The theorem will be proved in Section~5.
\bigskip\noindent
{\bf Remarks:}
1.
Theorem~\ref{smallMtheorem} provides better bound for the energy
difference than Theorem~\ref{generalMtheorem}.
Theorem~\ref{generalMtheorem}, on the other hand, covers much
wider range
of lowlying states.
We believe each theorem is optimal in each aspect.
We still do not know how to prove the optimal theorem which
interpolates
between the two theorems.
2.
The condition vi) for Theorem~\ref{smallMtheorem} can be weakened
to
$\gen\PL=0$ if one replaces the definition of the trial state
(\ref{Psi}) by
\begin{equation}
\PsL^{(M)}:=\frac{(\opplus)^M\PL+(\opminus)^M\PL}
{\norm{(\opplus)^M\PL+(\opminus)^M\PL}}.
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Lowlying states}
We shall discuss the implications of the above theorems
on the problem of
lowlying states in a sequence of finite systems.
The results for lowlying eigenstates have direct relevance to
numerical exact diagonalization approaches to quantum manybody
systems.
Take a sequence $\{\Lambda\}$ of finite lattices which tend to the
infinite lattice ${\bf Z}^d$.
For each $\Lambda$ (with $N$ sites) we consider a quantum
mechanical system on $\Lambda$ with the Hamiltonian
$\ham$, the generator $\gen$, the order operators
$(\opone,\optwo)$, and the ground state $\Phi_\Lambda$.
The corresponding eigenvalue of $\ham$ is denoted as $\EL^{(0)}$
(instead of $\EL$) since it is the ground state energy.
Let us now state precisely what we mean by lowlying states and
lowlying eigenstates in general.
\begin{definition}
A sequence of normalized
states $\{\PL'\}_\Lambda$ are called lowlying
states, if
\begin{equation}
\lim_\TDL\frac{1}{N}\rbk{(\PL',\ham\,\PL')\EL^{(0)}}=0
\label{LLS}
\end{equation}
holds, and if each $\PL'$ is orthogonal to the ground state $\PL$.
Lowlying states in which each state $\PL'$ happens to be
an eigenstate of the
Hamiltonian $\ham$ are called lowlying eigenstates.
\label{DEFlls}
\end{definition}
We assume that the sequence of models and the ground states
satisfy the assumptions i)iv) of the previous subsection
with constants $h$, $o$, $r$, and $\mu$
independent of $\Lambda$, or the assumptions i)vi) with these
constants and $\gamma$ independent of $\Lambda$.
Then Theorems~\ref{generalMtheorem} or \ref{smallMtheorem}
establish that $\{\PsL^{(M)}\}_\Lambda$,
constructed as (\ref{Psi}), with a fixed $M$ form
lowlying states.
By combining this observation with the variational argument, we
get rigorous control of lowlying eigenstates as follows.
\begin{coro}
Suppose that the sequence of models satisfy the assumptions
i)iv) with constants $h$, $o$, $r$, and $\mu$
independent of $\Lambda$.
Fix a lattice $\Lambda$ whose number of sites $N$ satisfies the bound
(\ref{N>}).
For each nonvanishing integer $M$ which satisfies
the
bound (\ref{M<}), one can find an eigenstate $\Phi_\Lambda^{(M)}$
of
the Hamiltonian $\ham$.
The state $\Phi_\Lambda^{(M)}$ is orthogonal to the ground state
$\Phi_\Lambda$, and the states $\Phi_\Lambda^{(M)}$ with
distinct $M$
are orthogonal to each other.
The energy eigenvalue $\EL^{(M)}$ of the state $\PL^{(M)}$
satisfies the bound
\begin{equation}
\frac{1}{N}(\EL^{(M)}\EL^{(0)})\le c_1\frac{M}{N},
\end{equation}
where $c_1$ is independent of $\Lambda$.
\end{coro}
\begin{proof}{Proof}
Since the ground state $\PL$ is an eigenstate of $\gen$, the
commutation relations (\ref{commutation2}) imply that the
variational states
$\PsL^{(M)}$ are orthogonal to the ground state
$\PL$.
Similarly we see that $\PsL^{(M)}$ with distinct $M$
are orthogonal to each other.
The desired result is then a consequence of the variational
principle
and Theorem~\ref{generalMtheorem}.%
\end{proof}
The above theorem establishes that the number of lowlying
eigenstates increases faster than any given small order of $N$.
As far as we know, this is the first time that the existence of ever
increasing numbers of lowlying eigenstates is proved.
We get a similar result from Theorem~\ref{smallMtheorem}.
\begin{coro}
Suppose that the sequence of models satisfy the assumptions
i)vi) with constants $h$, $o$, $r$, $\mu$, and $\gamma$
independent of $\Lambda$.
Fix a lattice $\Lambda$ with $N$ sites.
For each nonvanishing integer $M$
which satisfies the bound (\ref{M/N<}), one can find an eigenstate
$\Phi_\Lambda^{(M)}$ of the Hamiltonian $\ham$.
The state $\Phi_\Lambda^{(M)}$ is orthogonal to the ground state
$\Phi_\Lambda$, and the states $\Phi_\Lambda^{(M)}$ with
distinct $M$
are orthogonal to each other.
The energy eigenvalue $\EL^{(M)}$ of the state $\PL^{(M)}$
satisfies the bound
\begin{equation}
\frac{1}{N}(\EL^{(M)}\EL^{(0)})\le c_2\rbk{\frac{M}{N}}^2,
\label{llenergy}
\end{equation}
where $c_2$ is independent of $\Lambda$.
\end{coro}
\begin{proof}{Proof}
The same as the above.%
\end{proof}
We note that the lowlying eigenstates discussed here
have nothing to do with the NambuGoldstone excitations.
The finitesize counterparts of the NambuGoldstone
excitations should also have complicated structures, since
the (infinite volume) ground states themselves are reorganized
into lowlying states in a finite system
(as we shall see in Section~2.5).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{States with explicit symmetry breaking}
Here we restrict ourselves to models with $U(1)\times{\bf Z}_2$
symmetry, and discuss the implications of
Theorems~\ref{generalMtheorem}
and \ref{smallMtheorem} on the problem of symmetry breaking.
We shall see explicitly that, by forming linear combinations of the
ground state and the
lowlying states $\PsL^{(M)}$, we get states which
break the $U(1)$ symmetry explicitly.
The results discussed here is based on theorems by
Koma and Tasaki
\cite{KomaTasaki}.
Let us consider a sequence of models which satisfy the conditions
i)vi).
For a positive integer $k$, consider the state
\begin{equation}
\Xi^{(k)}_\Lambda:=\frac{1}{\sqrt{2k+1}}
\cbk{\PL+\sum_{M=1}^k
\rbk{\PsL^{(M)}+\PsL^{(M)}}},
\label{Xidef}
\end{equation}
where $\Lambda$ is taken sufficiently large so that the bounds
(\ref{N>}), (\ref{M/N<}) are valid for any $M$ with $M\le k$.
By using Theorem~\ref{generalMtheorem} or \ref{smallMtheorem}
and the fact that $(\PsL^{(i)},\ham\PsL^{(j)})=0$ for $i\ne j$,
we note that the state $\Xi^{(k)}_\Lambda$ is normalized,
and satisfies
\begin{equation}
\lim_\TDL \frac{1}{N}
\cbk{(\Xi^{(k)}_\Lambda,\ham\,\Xi^{(k)}_\Lambda)
\EL}=0,
\label{XihasE=0}
\end{equation}
for any $k$.
Thus $\{\Xi^{(k)}_\Lambda\}_\Lambda$ with a fixed $k$ form
lowlying states.
A remarkable properties of these lowlying states are the following.
\begin{theorem}
The expectation value of the order operators in the state
$\Xi^{(k)}_\Lambda$ satisfy
\begin{equation}
(\Xi^{(k)}_\Lambda,\optwo\,\Xi^{(k)}_\Lambda)=0,
\label{O2=0}
\end{equation}
and
\begin{equation}
\lim_{k\toinf}\lim_\TDL \frac{1}{N}
(\Xi^{(k)}_\Lambda,\opone\,\Xi^{(k)}_\Lambda)
\ge\sqrt{2}\mu o,
\label{XiOXi}
\end{equation}
where the prefactor $\sqrt{2}$ is modified if the model has a
higher symmetry.
For example, we replace $\sqrt{2}$
with $\sqrt{3}$
when
the model has an $SU(2)$ symmetry.
\label{KTtheorem}
\end{theorem}
\begin{proof}{Outline of proof}
The relation (\ref{O2=0}) follows by symmetry.
The relation (\ref{XiOXi}) is essentially proved in \cite{KomaTasaki}.
One only has to combine (7.26) of \cite{KomaTasaki} and Theorem~6.1
of \cite{KomaTasaki}.
respectively.
\end{proof}
The theorem establishes that the state $\Xi^{(k)}_\Lambda$
exhibits
explicit symmetry breaking.
By applying the $U(1)$ rotation $\exp[i\theta\gen]$ to the state
$\Xi^{(k)}_\Lambda$, we also get states in which the order
parameter is pointing different directions.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Infinite volume ground states}
Finally we shall discuss the relation between the lowlying
states in the sequence of finite systems and the
infinite volume ground states.
We again observe that the naive infinite volume limit of the finite
volume
ground states is not a pure state (hence is unphysical) when there is
an
obscured symmetry breaking.
By forming a linear combination of the (finitevolume)
ground state and the
lowlying states, and then taking an in
finite
volume limit, we get an infinite volume ground state
with explicit symmetry breaking, which state we
conjecture to
be pure.
In order to simplify the discussion, we make several assumptions
on
the model.
We assume that each finite lattice $\Lambda$ is a $d$dimensional
hypercubic lattice with periodic boundary conditions.
We assume that the Hamiltonian (\ref{ham})
and the order operators (\ref{orderOp}) are translation
invariant
in the sense that we can write
$h_x=\tau_x(h_o)$ and $o^{(\alpha)}_x=\tau_x(o^{(\alpha)}_o)$
for any $x$.
Here $\tau_x$ is the translation by the lattice
vector $x$ (which respects the periodic boundary conditions),
and the operators $h_o$ and $o_o$
are independent of $\Lambda$.
A local operator $A$ is an operator which acts nontrivially only on
a finite number of sites (or, more precisely, on a finite dimensional
Hilbert space $\bigotimes_{x\in{\cal S}(A)}{\cal H}_x$ with a finite
support set ${\cal S}(A)$).
Let
$\rho_\Lambda(\cdots)=
{\rm Tr}_{\calH_\Lambda}[(\cdots)\widetilde{\rho}_\Lambda]$
be a state of the system on $\Lambda$, where
$\widetilde{\rho}_\Lambda$ is an arbitrary density matrix
on $\calH_\Lambda$.
Given a sequence of (finitevolume) states
$\{\rho_\Lambda(\cdots)\}_\Lambda$, we (formally) define
\begin{equation}
\rho(A):=\lim_\TDL \rho_\Lambda(A)
\label{rholim}
\end{equation}
for each local operator $A$.
The above $\rho(\cdots)$ is a linear map from the space of
local
operators to the set of complex numbers ${\bf C}$.
We call $\rho(\cdots)$ a {\em state\/} of the infinite system.
(See Appendix~\ref{APgs} for the general definition of a state in an infinite
system.)
It might happen, however, that the limit (\ref{rholim})
does not exist for all local
$A$.
It is known that one can always choose a subsequence of
lattices so
that the limit is welldefined.
See Appendix~\ref{APgs} for a proof.
(An elementary proof can be constructed by using the
``diagonal sequence trick'', as is
illustrated, e.g., in Theorem~I.24 of \cite{ReedSimon}.)
We want to describe what we mean by a ground state of the infinite
system.
Since it is meaningless to talk about eigenstates or eigenvalues of
the total Hamiltonian $\ham$ when $\TDL$, a different point of
view is necessary.
Here we employ possibly the simplest definition for ground
states of an infinite system.
As we discuss in Appendix~\ref{APgs}, the present definition is
equivalent to the other definitions which are standard in
mathematical literature.
It simply says that a ground state should
minimize the
local energy.
\begin{definition}
We define the ground state energy density $\epsilon_0$ by
\begin{equation}
\epsilon_0:=\lim_\TDL
\inf_{\Phi_\Lambda\in{\cal H}_\Lambda}\frac{1}{N}
\bkt{\ham},
\label{epsilon0}
\end{equation}
where the limit always exists.
An infinite volume state $\omega(\cdots)$ is said to be a
ground state if it satisfies
\begin{equation}
\omega(h_x)=\epsilon_0,
\label{GS}
\end{equation}
for any $x\in{\bf Z}^d$.
\label{DEFgs}
\end{definition}
We also introduce the precise notion of {\em pure} states in an
infinite system.
We shall follow a physically natural definition which can be found
in Ruelle's textbook \cite{Ruelle}.
In short it says that a state in which any intensive bulk quantity
shows essentially no fluctuation is pure.
Since the requirement is believed to apply to any physically realizable
state of a large system, we might say that a translation invariant
state is physically natural if and only if it is pure in the following sense.
See also the remark at the end of the present section for the limitation
of the definition.
\begin{definition}
Let $\Omega$ be a hypercubic region in ${\bf Z}^d$, and denote the number of
sites in $\Omega$ by $\Omega$.
For an arbitrary local selfadjoint operator $A$, we define the corresponding
bulk quantity as
$A_\Omega:=\sum_{x\in\Omega}\tau_x(A)$,
where $\tau_x(A)$ is the translate of $A$ by a lattice vector $x$.
A translation invariant state $\rho(\cdots)$ is said to be {\em pure\/}
if, for any $A$, the intensive bulk quantity $A_\Omega/\Omega$
exhibits vanishing fluctuation in the sense that
\begin{equation}
\lim_{\Omega\toinf}\frac{1}{\Omega^2}
\rho\sbk{\rbk{A_\Omega\rho(A_\Omega)}^2}=0.
\end{equation}
\label{DEFpure}
\end{definition}
For each finite $\Lambda$, let $\PL$ be a ground state of
$\ham$.
We can assume $\PL$ is translation invariant.
Then it is easy to verify that the infinite volume state defined by
\begin{equation}
\omega(A):=\lim_\TDL \bkt{A},
\label{infGSseq}
\end{equation}
for any local operator $A$ (by taking a suitable subsequence),
is indeed an infinitevolume ground state in the sense of
Definition~\ref{DEFgs}.
Assume that each finite volume ground state $\PL$ exhibits
an obscured
symmetry breaking in the sense that it satisfies (\ref{LRO}) and
(\ref{NOSB}).
Then by using Lemma~\ref{flucLemma1}, we can show that
\begin{equation}
\frac{1}{\Omega^2}\omega
\rbk{(O^{(\alpha)}_\Omega\omega(O^{(\alpha)}_\Omega))^2}
\ge (\mu o)^2,
\label{omegaNOcluster}
\end{equation}
for any finite region $\Omega\subset{\bf Z}^d$,
where $O^{(\alpha)}_\Omega=\sum_{x\in\Omega}o^{(\alpha)}_x$.
This implies that the ground state $\omega(\cdots)$
is not a pure state, and is hence unphysical.
We still do not know how to construct a pure ground state in general.
In Appendix~\ref{APpure}, however, we present a general
construction of a pure
infinite volume ground state in a model where a {\em discrete}
symmetry is spontaneously broken and a gap above the first
lowlying eigenstate is generated.
In what follows, we make some observations which
suggest that the similar construction as in
Appendix~\ref{APpure} might work in models with a
broken continuous symmetry.
Let us start from a simple but important theorem
which summarizes the
relation between lowlying states and ground states of an
infinite system.
\begin{theorem}
Let $\{\PL'\}_\Lambda$ be lowlying states,
and assume that each $\PL'$ defines
translation invariant expectation values, i.e.,
$(\PL',A\,\PL')=(\PL',\tau_x(A)\,\PL')$ for any
$x\in\Lambda$ and for any local operator $A$.
Then the state
\begin{equation}
\omega'(\cdots):=\lim_\TDL(\PL',(\cdots)\,\PL'),
\end{equation}
defined by taking a suitable subsequence of lattices, is a ground
state.
\label{LLS=GS}
\end{theorem}
\begin{proof}{Proof}
The translation invariance implies
\begin{equation}
(\PL',\ham\,\PL')=N(\PL',h_x\,\PL'),
\end{equation}
for any $x\in\Lambda$.
Then the condition (\ref{LLS}) of lowlying states reads
\begin{equation}
\lim_\TDL\cbk{(\PL',h_x\,\PL')\bkt{h_x}}=0,
\end{equation}
which reduces to $\omega'(h_x)=\epsilon_0$ for any $x$.%
\end{proof}
It should be stressed that, in the above, the lowlying states $\PL'$
need
not be ground states or eigenstates of finite systems.
The states $\Xi_\Lambda^{(k)}$ defined in (\ref{Xidef}), and
its $U(1)$ rotations are lowlying states
with translation invariant expectation values.
Therefore Theorem~\ref{LLS=GS} implies that the limiting infinite
volume states
\begin{equation}
\omega_\theta(\cdots):=\lim_{k\toinf}\lim_\TDL
(e^{i\theta\gen}\,\Xi^{(k)}_\Lambda,(\cdots)\,
e^{i\theta\gen}\,\Xi^{(k)}_\Lambda),
\label{pure}
\end{equation}
defined by taking suitable subsequences, are ground states.
Theorem~\ref{KTtheorem}, on the other hand, implies that
\begin{equation}
\omega_\theta\sbk{o_x^{(1)}}=m\cos\theta,\quad
\omega_\theta\sbk{o_x^{(2)}}=m\sin\theta,
\label{pureorder}
\end{equation}
for any $x$, where $m\ge\sqrt{2}o\mu$ (or $m\ge\sqrt{3}o\mu$ if
the system has an $SU(2)$ symmetry).
The states $\omega_\theta(\cdots)$ are infinite volume
ground
states that explicitly shows a symmetry breaking.
It is believed that, in a system where a $U(1)$ symmetry
is spontaneously broken,
the nonpure ground state $\omega(\cdots)$
(defined in (\ref{infGSseq})) is decomposed as
\begin{equation}
\omega(\cdots)=\frac{1}{2\pi}\int_0^\theta d\theta\,
\omega^{\rm pure}_\theta(\cdots),
\label{puredeomp}
\end{equation}
where $\omega^{\rm pure}_\theta(\cdots)$ is a pure ground state
which satisfies (\ref{pureorder}) with $m$ replaced with some
$m_{\rm pure}$.
Assuming the decomposition (\ref{puredeomp}), we see,
for sufficiently large hypercubic region $\Omega$, that
\begin{equation}
\frac{1}{\Omega^2}\omega((O^{(1)}_\Omega)^2)
\simeq\frac{1}{2\pi}\int_0^\theta d\theta\,(m_{\rm pure}\cos\theta)^2
=\frac{(m_{\rm pure})^2}{2},
\label{sqrt2}
\end{equation}
where we have used that $\omega^{\rm pure}_\theta(\cdots)$
is pure.
Define the long range order parameter $\tilde{\mu}\ge0$ for the
state $\omega(\cdots)$ by
\begin{equation}
(\tilde{\mu}o)^2=\frac{1}{\Omega^2}
\lim_\TDL\omega((O^{(1)}_\Omega)^2).
\label{tildemu}
\end{equation}
Then (\ref{sqrt2}) implies the relation
$m_{\rm pure}=\sqrt{2}o\tilde{\mu}$.
Let us assume that the above $\tilde{\mu}$ is
equal to the largest
$\mu$ which satisfies (\ref{LRO}) for sufficiently large $N$.
(Unfortunately we can only prove the oneside bound
$\tilde{\mu}\ge\mu$ in general.
See Lemma~\ref{flucLemma1}.)
Then the relation $m_{\rm pure}=\sqrt{2}o\tilde{\mu}$
and the inequality $m\ge\sqrt{2}o\mu$ reduces to the
bound $m\ge m_{\rm pure}$.
Since the translation invariant pure ground states
are believed to have the largest
order parameter among all the ground states, this
suggests that we indeed have $m=m_{\rm pure}$.
This motivates us to further conjecture that the states
$\omega_\theta(\cdots)$ are nothing but the desired
{\em pure} ground states $\omega^{\rm pure}_\theta(\cdots)$.
(It is trivial to extend the present discussion to the case with a general
symmetry group.)
Unfortunately we have no direct evidences which support the
conjecture.
\bigskip\noindent
{\bf Remark:}
One should be aware that the notion of pure states for an infinite system is
distinct from that for ordinary quantum mechanics with finite degrees of
freedom.
(The two notions, of course, are closely related, especially if one
concentrates on the geometry of the space of all the states.)
>From Definition~\ref{DEFpure}, one finds, among other things,
that
pure states can be also characterized as ${\bf Z}^d$ergodic
states (see sections~6.3 and 6.5 of \cite{Ruelle}),
and that a nonpure translation invariant state can be uniquely
expressed as
an ``integral'' over pure states (see section 6.4 of \cite{Ruelle}).
A disadvantage in the above definition is that one has to
a priori assume the correct invariance of the states
in order to get a physically natural pure states.
See \cite{BratteliRobinson,Ruelle} for the notion of
pure states which does not rely on any invariance.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Examples}
In this section, we shall discuss some examples to
which our general results apply.
Although we only discuss selected models representing
typical situations, the reader can easily extend the following
analysis to much wider class of quantum manybody problems.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Ising model under transverse field}
We shall briefly discuss the Ising model under transverse magnetic
field with the Hamiltonian (\ref{Ising1}) considered in Section~1.2.
If the field $B$ is smaller than the critical value, the unique ground
state $\PL$ is expected to exhibit an ``obscured symmetry breaking''
in the sense that the relations $\bkt{\op}=0$ and
$\bkt{(\op)^2}\ge(m(B)N)^2$ hold, where the order operator $\op$ is
defined in (\ref{opIsing}).
This can be proved rigorously for sufficiently small $B$.
Then Horsch and von der Linden's theorem
(Theorem~\ref{M=1theorem}) ensures that there exists a
lowlying eigenstate whose excitation energy per site is bounded
from
above by a constant times $N^{2}$.
Note that the theorem does not reproduce the expected exponential
decay (\ref{expdecay}) of the energy difference.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Heisenberg antiferromagnet
with N\'eel order}
We discuss the Heisenberg quantum antiferromagnetic spin system,
which is a typical model with a spontaneously broken continuous
symmetry.
Let $\Lambda$ denote the
$d$dimensional $L\times\cdots\times L$ hypercubic lattice with
periodic boundary conditions, where $L$ is an even integer.
With each site $x\in\Lambda$, we associate
the spin operators $(S^{(1)}_x,S^{(2)}_x,S^{(3)}_x)$
for spin $S=1/2,1,3/2,\ldots$.
The Hamiltonian (\ref{ham}) is defined by the local Hamiltonian
\begin{equation}
h_x = \frac{1}{2}\sum_{y;xy=1}
\rbk{S^{(1)}_xS^{(1)}_y+S^{(2)}_xS^{(2)}_y+\lambda
S^{(3)}_xS^{(3)}_y},
\label{HAF}
\end{equation}
where $0\le\lambda\le1$, and the sum is over
the sites $y$ neighboring to $x$.
When $L$ is finite, the ground state $\PL$ of the
Hamiltonian
(\ref{HAF}) is rigorously known \cite{Marshall,LiebMattis,AffleckLieb}
to be unique and
satisfies
\begin{equation}
C_\Lambda\PL=0,
\end{equation}
with $C_\Lambda=\sum_{x\in\Lambda}S^{(3)}_x$.
For $\alpha=1,2$, we define the order operators (\ref{orderOp}) by
the local order operators
\begin{equation}
o^{(\alpha)}_x=\left\{\begin{array}{ll}
S^{(\alpha)}_x&\mbox{if $x\in A$}\\
S^{(\alpha)}_x&\mbox{if $x\in B$},
\end{array}\right.
\label{orderHAF}
\end{equation}
where we have decomposed $\Lambda$ into two sublattices as
$\Lambda=A\cup B$ so that for any neighboring sites $x,y$ we have
either $x\in A$, $y\in B$ or $x\in B$, $y\in A$.
It is expected, and is partially proved by the DysonLiebSimon
method and its extensions
\cite{DysonLiebSimon,Jordao,KennedyLiebShastry1,%
KennedyLiebShastry2,Kubo,KuboKishi,NishimoriKubo,%
NishimoriOzeki,Ozeki}
that, for any
$d\geq2$ and $0\le\lambda\le1$, the ground state $\Phi_\Lambda$
exhibits a
N\'eeltype long range order.
We expect that the condition (\ref{LRO}) is valid with $\mu>0$
(where $\mu$ depends on $d$ and $\lambda$).
On the other hand, the absence of explicit symmetry breaking as in
(\ref{NOSB}) is obvious from the uniqueness of the ground state.
We thus conclude that there is an obscured symmetry breaking.
Assuming the existence of the N\'eel order (\ref{LRO}),
we can apply our lowlying
states theorems.
The model has a desired $U(1)\times{\bf Z}_2$ symmetry.
We can use
Theorems~\ref{generalMtheorem} and \ref{smallMtheorem} by noting
that the present order operators $\op^{(1)}$ and $\op^{(2)}$, along
with the above defined
$\gen$, satisfy the requirements in
the theorems with $\gamma=1$.
Then we can make use of the general considerations in Section~2.3,
and conclude that the number of independent lowlying eigenstates
increases faster than any given small order of $N$.
For the $SU(2)$ invariant Heisenberg antiferromagnet with
$\lambda=1$ in (\ref{HAF}), Momoi \cite{Momoi} recently succeeded in
improving the present result.
In order to apply the general results in Section~2.5, we need
an extra care.
Since the order operators (\ref{orderHAF}) do not satisfy the
requirement of the translation invariance, we have to redefine
what we mean by a ``site''.
We group together $2^d$ sites forming a $2\times\cdots\times2$
hypercubic region (i.e., a unit cell), and call the group a ``site''.
After redefining the local Hilbert space, the local Hamiltonian,
and the local order operators according to the new notion of
``sites'', the model satisfies the assumptions of Section~2.5.
We can then form (presumably pure) ground states with explicit
symmetry breaking as
in (\ref{pure}).
We stress that the applicability of our lowlying states theorems is
not limited to models on the hypercubic lattice.
For example, the model on the triangular lattice with the same
Hamiltonian (\ref{HAF}) with $0\le\lambda\le1$, where $y$ is
summed over nearest neighbor sites of $x$, has been attracting
considerable
interest.
(See \cite{Bernu,Azaria,Leung} and many early references therein.)
Anticipating the socalled $120^\circ$ structure, we set the order
operators as
\begin{equation}
\op^{(1)}=
\sum_{x\in A}
S^{(1)}_x
+\sum_{x\in B}
\rbk{\frac{1}{2}S^{(1)}_x\frac{\sqrt{3}}{2}S^{(2)}_x}
+\sum_{x\in C}
\rbk{\frac{1}{2}S^{(1)}_x+\frac{\sqrt{3}}{2}S^{(2)}_x},
\end{equation}
\begin{equation}
\op^{(2)}=
\sum_{x\in A}
S^{(2)}_x
+\sum_{x\in B}
\rbk{\frac{\sqrt{3}}{2}S^{(1)}_x\frac{1}{2}S^{(2)}_x}
+\sum_{x\in C}
\rbk{\frac{\sqrt{3}}{2}S^{(1)}_x\frac{1}{2}S^{(2)}_x},
\end{equation}
where we have divided the triangular lattice into three sublattices
$A$, $B$, and $C$, so that neighboring sites $x,y$ always belong to
different sublattices.
These order operators again satisfy the conditions for the
theorems with the generator $\gen=\sum_xS^{(3)}_x$, and
Theorems~\ref{generalMtheorem} and \ref{smallMtheorem} are
applicable.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{BoseEinstein condensation in hard core Bose gas on
lattice}
We shall give a brief discussion on the BoseEinstein condensation problem.
It turns out that, by following the general discussions given in the
previous section, we are naturally led to consider ground states
with unconserved particle number.
Let $\Lambda$ be the $d$dimensional $L\times\cdots\times L$
hypercubic
lattice with periodic boundary conditions, where $L$ is an even
integer, and
$d\ge2$.
With each site $x$, we associate the creation operator $a^*_x$ and
the
annihilation operator $a_x$ of a spinless boson.
We consider the Hamiltonian (\ref{ham}) defined by
\begin{equation}
h_x = \frac{K}{2}\sum_{y;xy=1}(a^*_xa_y+a^*_ya_x)
+Vn_x(n_x1),
\label{hamBose}
\end{equation}
where the sum runs over the sites neighboring to $x$, and
$n_x=a^*_xa_x$ denotes the number operator.
We shall take the limit of infinitely
large onsite repulsion $V\toinf$
before
the infinite volume limit, and restrict ourselves to the states with
finite
energies (in a finite volume).
This defines the socalled hard core Bose gas.
It is wellknown that the hard core Bose gas on a lattice is
equivalent to the
$S=1/2$ quantum XY model on the same lattice
\cite{MatsubaraMatsuda}.
Based on the equivalence and an extension of the infrared bound
method of
Dyson, Lieb and Simon \cite{DysonLiebSimon}, it was proved by
Kennedy, Lieb and Shastry \cite{KennedyLiebShastry2},
and by
Kubo and Kishi \cite{KuboKishi}
that the present model exhibits a BoseEinstein condensation
in the
following sense.
Let $\PL$ be the unique ground state of the Hamiltonian
(\ref{hamBose}) with
the particle number equal to $N=L^d/2$.
Define the order operators (\ref{orderOp}) by
\begin{equation}
o^{(1)}_x=\calP\frac{a^*_x+a_x}{2}\calP,\quad
o^{(2)}_x=\calP\frac{a^*_xa_x}{2i}\calP,
\label{boseOrder}
\end{equation}
where $\calP$ is the projection operator onto the space of finite energy
states, i.e., $\Phi$ such that $n_x(1n_x)\Phi=0$ for any $x$.
Then the result of \cite{KennedyLiebShastry2,KuboKishi} is that the
condition
of the long range order (\ref{LRO}) holds with a finite $\mu$.
On the other hand the absence of explicit symmetry breaking
as in (\ref{NOSB}) is
manifest since the state $\PL$ has a fixed particle number.
We see that there is an obscured symmetry breaking.
It is not hard to see that we can apply our
Theorems~\ref{generalMtheorem} and \ref{smallMtheorem} to this
situation.
For this, we replace the Hamiltonian (\ref{hamBose}) with
$\calP h_x\calP$.
Note that the latter is a bounded operator, while the former is
unbounded.
Since we are only dealing with states with finite energies,
this replacement does not change any physics.
By using the replaced Hamiltonian and the order operators
(\ref{boseOrder}), we find that the model has a
desired $U(1)\times{\bf Z}_2$ symmetry.
The relevant $U(1)$ symmetry is that for the quantum mechanical
phase
generated by $\gen=\sum_{x\in\Lambda}(n_x1/2)$,
and the ${\bf Z}_2$ symmetry is the holeparticle symmetry.
The general discussions in Section~2 implies that one has lowlying
states in
the sectors with different particle numbers, and the (candidate of)
pure
infinite volume ground state $\omega_\theta(\cdots)$ is
constructed as in
(\ref{pure}).
The corresponding lowlying state (\ref{Xidef}), in this situation,
becomes
\begin{equation}
\Xi^{(k)}_\Lambda=\frac{1}{\sqrt{2k+1}}\cbk{
\PL+\sum_{M=1}^k\rbk{
\frac{\calP(\sum_{x\in\Lambda}a^*_x)^M\PL}
{\norm{\calP(\sum_{x\in\Lambda}a^*_x)^M\PL}}
+\frac{(\sum_{x\in\Lambda}a_x)^M\PL}
{\norm{(\sum_{x\in\Lambda}a_x)^M\PL}}
}}.
\label{XiforBose}
\end{equation}
A remarkable feature of the (presumably pure) ground state
$\omega_\theta(\cdots)$
is that it is constructed by summing up the states with different
particle
numbers as in (\ref{XiforBose}).
We further find from (\ref{pureorder}) that the state has
nonvanishing
expectation values of the creation and annihilation operators,
for example, as
\begin{equation}
\omega_{\theta=0}(a^*_x)=\omega_{\theta=0}(a_x)\ge\sqrt{2}o\mu,
\end{equation}
for any $x$.
In a theoretical treatment of the BoseEinstein condensation, it is
standard to
consider states without particle number conservation and
with nonvanishing
expectation values for creation and annihilation operators.
Usually such states are introduced within the framework of certain meanfield
theory.
We have seen that such states arise naturally if one tries to
consider pure
infinite volume ground states.
\bigskip\noindent
{\bf Remark:}
Since it is physically meaningless to compare the energies of two
states with
different particle numbers, the existence of lowlying states in the
present
situation has less physical significance.
A more important fact is that the states $\omega_\theta(\cdots)$
are really
infinite volume ground states.
This point requires further discussions.
A physically natural setup in the present problem
is to consider a finite system with a
fixed particle
number.
Then one can add an extra term $\nu\sum_{x\in\Lambda}n_x$
to the Hamiltonian without
changing any physics.
It is clear that the definition of infinite volume ground states
employed in
Section~2.5 is sensitive to the value of the ``chemical potential''
$\nu$,
once we relax the constraint of the particle number conservation.
Better definitions in the present situation are those of $\calG_1$
or
$\calG_2$ in Appendix~\ref{APgs}, with allowed perturbations (local operators
$A$ in
the former, and maps $T$ in the latter) restricted only to those
preserve the
particle number.
This definition is clearly independent of the value of $\nu$.
To prove that $\omega_\theta(\cdots)$ is a ground state in this
sense, it
suffices to use the relations between different definitions
(Proposition~\ref{G3}) along with the fact that
$\omega_\theta(\cdots)$ is a
ground state (in the sense of Section~2.5 or $\calG_3$) when
$\nu=0$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Superconductivity in lattice electron systems}
We shall discuss applications of our theorems to
lattice electron problems.
A class of possible applications deal with magnetic ordering in an
electron
model.
Since such problems can be treated in exactly the same manner as
the quantum spin systems discussed previously, we leave the
details to interested readers.
We shall concentrate on a symmetry breaking intrinsic to
interacting
electron systems, namely, superconductivity.
Consider an electron system on a finite lattice $\Lambda$,
and denote by $ c^*_{x\sigma}$ and $ c_{x\sigma}$ the creation and
annihilation
operators, respectively, of an electron at site $x$ with spin
$\sigma=\uparrow,\downarrow$.
We consider a Hamiltonian which commutes with the total electron
number
\begin{equation}
N_{\rm e}=\sum_x n_{x\uparrow}+n_{x\downarrow},
\end{equation}
where $n_{x\sigma}= c^*_{x\sigma} c_{x\sigma}$.
A typical example is the socalled Hubbard model.
(See, for example, \cite{Lieb,Montorsi}.)
A class of Hubbard models with attractive interactions are believed
to exhibit superconductivity in their ground states.
It is also expected that certain Hubbard models with repulsive
interaction also exhibit superconductivity.
The latter possibility is interesting not only because of its possible
connection with high$T_{\rm c}$ superconductivity, but as a new
type of collective phenomena in strongly interacting electron
systems.
A standard superconducting phase can be characterized by a
condensation of certain electron pairs, which manifests itself as a
long range order in electron pairing correlation function.
For example the condensation of singlet pairs can
be measured as a long range order (\ref{LRO}) with respect to the
order operators defined by
\begin{equation}
o_x^{(1)} =
\frac{1 }{2}\rbk{p_xc^*_{x\uparrow}c^*_{x\downarrow}
\overline{p_x}c_{x\uparrow}c_{x\downarrow}},
\end{equation}
\begin{equation}
o_x^{(2)} =
\frac{1}{2i}\rbk{p_xc^*_{x\uparrow}c^*_{x\downarrow}+
\overline{p_x}c_{x\uparrow}c_{x\downarrow}},
\end{equation}
where $p_x$ (with $p_x=1$) is a certain phase factor.
It is easily checked that the model has a $U(1)$ symmetry and
satisfies the conditions for
Theorem~\ref{generalMtheorem} with $\gen =(N_{\rm e}N)/2$,
where $N$ denotes the number of sites in $\Lambda$.
The theorem states that the existence of an
electron pair condensation inevitably leads to the appearance of
lowlying eigenstates whose number increase faster than any
given
small order of the system size $N$.
Except for halffilled models with special Hamiltonians, the
models do not have a $U(1)\times{\bf Z}_2$ symmetry necessary to
apply Theorem~\ref{smallMtheorem}.
We remark that it is possible to treat other types of pairing with
some extra care.
To treat triplet pairing, for example, we first redefine what we
mean by sites of the lattice.
We divide the lattice $\Lambda$ into a disjoint union of
nonoverlapping pairs of sites.
We then regard each pair $\{x,y\}$ as a ``site'' of the lattice.
The local order parameters to measure a possible condensation
of triplet pairs are defined by summing up the following local
order operators over all the ``sites''.
\begin{equation}
o_{\{x,y\}}^{(1)} =
\frac{1}{2}\rbk{c^*_{x\uparrow}c^*_{y\downarrow}
+c^*_{x\downarrow}c^*_{y\uparrow}
c_{x\uparrow}c_{y\downarrow}
c_{x\downarrow}c_{y\uparrow}},
\end{equation}
\begin{equation}
o_{\{x,y\}}^{(2)} =
\frac{1}{2i}\rbk{c^*_{x\uparrow}c^*_{y\downarrow}
+c^*_{x\downarrow}c^*_{y\uparrow}
+c_{x\uparrow}c_{y\downarrow}
+c_{x\downarrow}c_{y\uparrow}}
\end{equation}
We again set $\gen =(N_{\rm e}N)/2$, and apply
Theorem~\ref{generalMtheorem} to control lowlying states.
Although there are no nontrivial electron system in which the
existence of a superconducting phase is proved, we believe our
results have potential usefulness, for example, in future numerical
works.
\bigskip\noindent
{\bf Remark:}
The lattice fermion problems considered here are different from
other examples in that the corresponding Hilbert space is not a
simple tensor product of local Hilbert spaces as in (\ref{Hilb})
or (\ref{Hilb2}).
This difference causes no problem for proving our lowlying
states theorems since we only make use of some
commutation relations between operators in the proof.
But some results about infinite volume states, which are mainly
quoted from literature in Section~2.5 and Appendix~\ref{APgs},
may not apply.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{$S=1$ antiferromagnetic chain}
A rather interesting application of the lowlying states theorem
can be found in the problem related
to the socalled Haldane gap.
Let $\Lambda$ be the onedimensional open chain $\{1,2,\ldots,N\}$.
With each site $x\in\Lambda$, we associate the three dimensional
Hilbert space for an $S=1$ quantum spin, and denote by
$(S^{(1)}_x,S^{(2)}_x,S^{(3)}_x)$ the corresponding spin operators.
We consider the Hamiltonian
\begin{equation}
H_\Lambda = \sum_{x=1}^{N1}
\rbk{S^{(1)}_xS^{(1)}_{x+1}+S^{(2)}_xS^{(2)}_{x+1}+\lambda
S^{(3)}_xS^{(3)}_{x+1}}
+D\sum_{x=1}^N (S^{(3)}_x)^2,
\label{S=1Ham}
\end{equation}
where $\lambda$ and $D$ are parameters.
Haldane \cite{Haldane} argued that, in a
finite range of
the parameter space including the Heisenberg point $\lambda=1$,
$D=0$, the model is in an exotic phase (now called
the ``Haldane phase'') where the unique infinite volume
ground state is accompanied by a finite excitation gap.
This was quite surprising since the Heisenberg antiferromagnetic
chain with $S=1/2$ is known to have vanishing gap from the Bethe
ansatz solution.
(See also \cite{AffleckLieb}.)
Haldane's prediction was that the gapful Haldane phase exists if and
only if the spin $S$ is an integer.
The existence of the Haldane phase in $S=1$ chains has been
proved rigorously only in the exactly solvable VBS model
\cite{AffleckKennedy}, its
non$SU(2)$invariant extensions \cite{Fannes}, and perturbations to
the dimerized VBS model \cite{KennedyTasaki}.
The ref. \cite{Fannes} contains a general treatment of the VBStype
models, and the $S=1$ model mentioned here is one of the examples.
The $S=1$ model of \cite{Fannes} and
their method of constructing the ground state
(but not the proof of the existence of a gap)
were rediscovered by other authors \cite{Klumper}.
The ground state in the Haldane phase is disordered in the sense that
the spinspin correlation functions decay exponentially.
Den~Nijs and Rommelse \cite{denNijs} pointed out that the
ground state in the Haldane phase of an $S=1$ chain has a ``hidden
antiferromagnetic
order''.
For $i=1,2,3$, let the string order operator be
\begin{equation}
\op^{(i)}:=\sum_{x=1}^N S^{(i)}_x\exp[i\pi\sum_{y=1}^{x1}S^{(i)}_y].
\label{stringoperator}
\end{equation}
If we denote the unique normalized ground state
for finite $\Lambda$ with $N$
sites as $\PL$, we expect to have
\begin{equation}
\bkt{(\op^{(i)})^2}\ge(\sigma^{(i)} N)^2,
\label{stringorder}
\end{equation}
in the Haldane phase with $\sigma^{(1)}=\sigma^{(2)}>0$ and
$\sigma^{(3)}>0$.
The condition (\ref{stringorder}) corresponds to the
antiferromagnetic ordering of spins with $S^{(i)}_x=1$ and
$S^{(i)}_x=1$, where spins with $S^{(i)}_x=0$ are inserted randomly
in
between them \cite{denNijs,Tasaki}.
Again the existence of the ``hidden antiferromagnetic order'' has
been established only for special classes of models mentioned above.
Let us consider the following trial states
\begin{equation}
\PsL^{(i)}=\frac{\op^{(i)}\PL}{\norm{\op^{(i)}\PL}},
\label{Ktrip}
\end{equation}
for each $i=1,2,3$.
We also introduce the operators
\begin{equation}
U_\Lambda^{(i)}=\exp[i\pi\sum_{x=1}^NS^{(i)}_x],
\end{equation}
which rotates all the spins by $\pi$ around the $i$th axis.
Note that the unique ground state satisfies
$U_\Lambda^{(i)}\PL=\PL$ for any $i$, and the trial states
(\ref{Ktrip}) satisfy
\begin{equation}
U_\Lambda^{(i)}\PsL^{(j)}=\left\{
\begin{array}{ll}
\PsL^{(j)}&\mbox{if $i=j$},\\
\PsL^{(j)}&\mbox{if $i\ne j$}.
\end{array}
\right.
\end{equation}
>From the difference of parities, we find that the four states $\PL,
\PsL^{(1)}, \PsL^{(2)}$, and $\PsL^{(3)}$ are orthogonal to each other.
It is not hard to check that we can apply Horsch and von der Linden's
theorem (Theorem~\ref{M=1theorem})
to this situation.
We find, for each $i=1,2,3$, that the trial states
$\PsL^{(i)}$ are lowlying states.
The Hamiltonian (\ref{S=1Ham}) on a finite open chain should
have (at least) three lowlying eigenstates.
(Again the excitation energies of the lowlying eigenstates are
believed
to decay exponentially in $N$, but the bound in the theorem fails to
reproduce this.)
These lowlying eigenstates are nothing but the socalled ``Kennedy
triplet'' which have been
observed in exact solutions \cite{AffleckKennedy}, in numerical simulations
\cite{Kennedy} and
in actual experiments in impurity doped samples \cite{Hagiwara}.
The existence of the Kennedy triplet is characteristic in the Haldane
gap system on a finite open chain.
In a periodic chain, it is believed that the unique ground state is
accompanied by a finite excitation gap, and there are no lowlying
states.
Fundamental connection between the hidden order
(\ref{stringorder}) and the existence
of the lowlying triplet was discussed by Kennedy and Tasaki
\cite{KennedyTasaki} from the view point of the
``hidden ${\bf Z}_2\times{\bf Z}_2$ symmetry breaking''.
Our remark here is that this connection can be made (formally)
explicit at least in one direction\footnote{
That the existence of a string order should imply the existence of
the lowlying triplet was pointed out to one of the authors (H.T.)
by Ian Affleck in July 1992.
The present work initially emerged from an attempt to
look for a proof of his claim, although the main interest
of the authors has shifted in the long run to problems with
continuous symmetry breaking.
}.
The present example is different from the others in that the three
lowlying states and the unique (finite volume) ground state
converge to a unique infinite volume ground state.
This is related to the nonlocal nature of the order operators.
See \cite{AffleckKennedy,Kennedy,KennedyTasaki} for more details.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Proof of first theorem}
In the present section, we prove Theorem~\ref{generalMtheorem} for
$M>0$.
Throughout the proof, we drop the subscript $\Lambda$ from
$\op^\pm, \ham,
E_\Lambda, \PL, \PsL^{(M)}$, etc.
Our goal is to bound the quantity
\begin{eqnarray}
\Delta^{(M)}
&:=&
\frac{1}{N}\cbk{(\Psi^{(M)},H\,\Psi^{(M)})E}
\ret
&=&
\frac{\gs{\kminus^M H \kplus^M}\gs{\kminus^M \kplus^M H}}
{N\gs{\kminus^M \kplus^M}}
\ret
&=&
\sum_{x\in\Lambda}
\frac{\gs{\kminus^M[h_x,\kplus^M]}}
{N\gs{\kminus^M \kplus^M}}.
\label{DeltaDef}
\end{eqnarray}
The final expression in (\ref{DeltaDef}) motivates us to decompose
the
operator $\Oplus$ as
\begin{equation}
\Oplus = \Qx+\Rx,
\end{equation}
with
\begin{equation}
\Qx:=\sum_{y\not\in\calS_x}o^+_y,\quad
\Rx:=\sum_{y\in\calS_x}o^+_y.
\end{equation}
Note that we have $[\Qx,h_x]=0$ and $[\Qx,\Rx]=0$ from the
assumptions ii)
and i), respectively.
Roughly speaking, $\Qx$ is the main part of the operator $O^+$, and
is
expected to behave almost similarly to $O^+$.
This is the basic idea of the present proof.
The following lemma makes the idea into a precise estimate.
\begin{lemma}
Suppose that the conditions (\ref{N>}) and (\ref{M<}) for $N$ and $M$
are
satisfied.
Then for $k=1,2,\ldots,M$, we have
\begin{equation}
\frac{\gs{\KQsx^{Mk}\KQx^{Mk}}}
{\gs{\KQsx^M\KQx^M}}
\le (\mu o N)^{2k}.
\label{QQBound}
\end{equation}
\label{QQLemma}
\end{lemma}
We shall prove the lemma at the end of the present section.
By using the expansion formula
\begin{equation}
\kplus^M=\sum_{k=0}^M\Mk\KQx^{Mk}\KRx^k,
\label{OExp}
\end{equation}
we get
\begin{eqnarray}
&&\gs{\kminus^M[h_x,\kplus^M]}
\ret
&=&
\sum_{k=0}^M\sum_{\ell=0}^M\Mk\Ml
\gs{\KQsx^{Mk}\KRsx^k[h_x,\KQx^{M\ell}\KRx^\ell]}
\ret
&=&
\sum_{k=0}^M\sum_{\ell=1}^M\Mk\Ml
\gs{\KQsx^{Mk}\KRsx^k[h_x,\KRx^\ell]\KQx^{M\ell}}.
\label{OhO1}
\end{eqnarray}
The Schwartz inequality and the definition (\ref{DEFnorm}) of the
operator
norm yield the following useful bounds
\begin{eqnarray}
\abs{\gs{A^*BC}}
&\le&
\sqrt{\gs{A^*A}\gs{C^*B^*BC}}
\ret
&\le&
\norm{B}\sqrt{\gs{A^*A}\gs{C^*C}},
\label{ABC}
\end{eqnarray}
for general operators $A$, $B$, and $C$.
By applying (\ref{ABC}) to (\ref{OhO1}) and noting that ii) and iii)
imply
$\norm{\KRsx^k[h_x,\KRx^\ell]}\le 2h(2ro)^{k+\ell}$,
we get
\begin{eqnarray}
&&\abs{\frac{\gs{\kminus^M[h_x,\kplus^M]}}{\gs{\KQsx^M\KQx^M}}}
\ret
&\le&
\sum_{k=0}^M\sum_{\ell=1}^M\Mk\Ml2h(2ro)^{k+\ell}
\frac{\sqrt{\gs{\KQsx^{Mk}\KQx^{Mk}}\gs{\KQsx^{M\ell}\KQx^{M
\ell}}}}
{\gs{\KQsx^{M}\KQx^{M}}}
\ret
&\le&
2h\sum_{k=0}^M\sum_{\ell=1}^M\Mk\Ml(2ro)^{k+\ell}(\mu o N)^{
(k+\ell)}
\ret
&=&
2h\rbk{1+\frac{2r}{\mu N}}^M\cbk{\rbk{1+\frac{2r}{\mu N}}^M1}
\ret
&\le&
2h \exp\sbk{\frac{2r}{\mu}\frac{M}{N}}
\cbk{\exp\sbk{\frac{2r}{\mu}\frac{M}{N}}1}
\ret
&\le&
2h e^{\mu/4}\frac{8r(e^{\mu/4}1)}{\mu^2}\frac{M}{N}
\ret
&=&
16rh\mu^{2}(e^{\mu/2}e^{\mu/4})\frac{M}{N},
\label{OhO2}
\end{eqnarray}
where we have used the bounds (\ref{QQBound}) and (\ref{M<}).
Again using (\ref{OExp}), we get
\begin{eqnarray}
&&\gs{\kminus^M\kplus^M}
\ret&=&
\gs{\KQsx^M\KQx^M}
\ret&+&
{\sum_{k,\ell}}'\Mk\Ml\gs{\KQsx^{Mk}\KRsx^k\KRx^\ell\KQx^{M
\ell}},
\end{eqnarray}
where the summation in the righthand side runs over all
$k,\ell=0,1,\ldots,M$ except for $k=\ell=0$.
>From (\ref{ABC}) and (\ref{QQBound}), we get
\begin{eqnarray}
&&\abs{\frac{\gs{\kminus^M\kplus^M}}{\gs{\KQsx^M\KQx^M}}}
\ret
&\ge&
1{\sum_{k,\ell}}'\Mk\Ml(2ro)^{k+\ell}
\frac{\sqrt{\gs{\KQsx^{Mk}\KQx^{Mk}}\gs{\KQsx^{M\ell}\KQx^{M
\ell}}}}
{\gs{\KQsx^M\KQx^M}}
\ret
&\ge&
1\cbk{\rbk{1+\frac{2r}{\mu N}}^{2M}1}
\ret
&\ge&
2e^{\mu/2}.
\label{OOQQ}
\end{eqnarray}
Note that, since $0\le\mu\le1$, we have $2e^{\mu/2}\ge2
\sqrt{e}>0$.
By combining (\ref{DeltaDef}), (\ref{OhO2}), and (\ref{OOQQ}), we
finally get
\begin{equation}
\abs{\Delta^{(M)}}
\le
16rh\frac{e^{\mu/2}e^{\mu/4}}{\mu^2(2e^{\mu/2})}\frac{M}{N}
=c_1\frac{M}{N}.
\end{equation}
\begin{proof}{Proof of Lemma~\ref{QQLemma}}
We write $a_m:=\gs{\KQsx^m\KQx^m}$.
We will prove that for $m=1,2,\ldots,M$, we have
\begin{equation}
\frac{a_m}{a_{m1}}\ge(\mu o N)^2.
\label{a/a}
\end{equation}
Then the desired bound (\ref{QQBound}) follows by multiplying
(\ref{a/a})
with $m=Mk+1,Mk+2,\ldots,M$.
We start by evaluating $a_1$ as
\begin{eqnarray}
a_1 &=& \gs{(\Ominus\Rsx)(\Oplus\Rx)}
\ret
&\ge& \gs{\Ominus\Oplus}2(2o)^2rN
\ret
&=& \frac{1}{2}\cbk{\gs{\Ominus\Oplus}+\gs{\Oplus\Ominus}
+\gs{[\Ominus,\Oplus]}}8o^2rN
\ret
&\ge& \gs{\kone^2}+\gs{(O^{(2)})^2}2o^2(1+4r^2)N,
\end{eqnarray}
where we have used i), ii), iii) to bound the norm of the
commutators.
By substituting the assumption (\ref{LRO}) on the existence of a long
range
order, and the bound (\ref{N>}) for $N$, we get
\begin{eqnarray}
a_1&\ge&2(\mu o N)^2 \cbk{1\frac{1+4r^2}{\mu^2N}}
\ret
&\ge&2(\mu o N)^2 \cbk{1\frac{1+4r^2}{16r^2}}.
\label{a1Bound}
\end{eqnarray}
Since $r\ge2$, we have shown that $a_1>0$.
Next we use the Schwartz inequality to get
\begin{eqnarray}
(a_{m1})^2 &=& \gs{\KQsx^{m2}\Qsx\KQx^{m1}}^2
\ret
&\le& \gs{\KQsx^{m2}\KQx^{m2}}\gs{\KQsx^{m1}\Qx\Qsx\KQx^{m
1}}
\ret
&=&
a_{m2}\cbk{\gs{\KQsx^m\KQx^m}+\gs{\KQsx^{m
1}[\Qx,\Qsx]\KQx^{m1}}}
\ret
&\le&
a_{m2}\cbk{a_m+4o^2Na_{m1}},
\label{am1}
\end{eqnarray}
where the final inequality follows from (\ref{ABC}).
Assuming that $a_{m2}\ne0$ and $a_{m1}\ne0$ (which is true for
$m=2$),
we find from (\ref{am1}) that
\begin{equation}
\frac{a_m}{a_{m1}}\ge\frac{a_{m1}}{a_{m2}}4o^2N.
\label{recBound}
\end{equation}
The rest of the proof is easy.
Assume that, for a fixed $m$, $a_{m'}\ne0$ for all $m'0$, proves inductively
$\gs{\kone^{2m}}>0$ for any
$m$.
By rearranging (\ref{OO1}), we get
\begin{equation}
\frac{\gs{\kone^{2(m1)}}}{\gs{\kone^{2m}}}
\le \frac{\gs{\kone^{2(m2)}}}{\gs{\kone^{2(m1)}}}
\le\cdots\le
\frac{1}{\gs{\kone^2}}\le\frac{1}{(\mu oN)^2},
\end{equation}
where we used (\ref{LRO}).
We also note that the definition (\ref{DEFnorm}) implies
\begin{equation}
\gs{\kone^{2m}}\le\norm{\oone}^2\gs{\kone^{2(m1)}}
\le(oN)^2\gs{\kone^{2(m1)}}.
\label{OO2}
\end{equation}
By substituting (\ref{OO1}) and (\ref{OO2}) into
\begin{equation}
\frac{b_{m1}}{b_m}=
\frac{2m(2m1)}{(2m)^2}\frac{\gs{\kone^{2(m
1)}}}{\gs{\kone^{2m}}},
\end{equation}
which follows from the definition (\ref{bmDef}), we get
(\ref{b/b}).%
\end{proof}
One of the main ingredients in the proof is the following
Lemma~\ref{hardLemma}, which allows us to approximate
expectation
values including $O^\pm$ with those including the selfadjoint
operator
$\oone$.
Let $A$ be an operator written as $A=\sum_{x\in\Lambda}a_x$,
where $[a_x, o_y^{(\alpha)}]=0$ for
$\alpha=1,2$ if $y\not\in\calS_x$, and
$\norm{a_x}\le a$ with a finite constant $a$ for any $x$.
The support set $\calS_x$ is the same for that of $h_x$.
In the following we set $\osig{}=\Oplus$ or $\Ominus$ depending on
$\sigma=+1$ or $1$.
\begin{lemma}
Let $K,L$ be nonnegative integers which satisfy
\begin{equation}
\frac{48r}{\mu^2}\frac{K+L}{N}
+\frac{3\gamma}{2o^2\mu^2}\frac{(K+L)^3}{N^2} \le 1.
\label{K+Lcond}
\end{equation}
We assume that $A$ satisfies
\begin{equation}
[C,\kplus^{KL}A]=0,
\label{Acond}
\end{equation}
where $C$ is the generator of the $U(1)$ symmetry.
(For $KL<0$, we set $\kplus^{KL}=\kminus^{LK}$.)
The relation (\ref{Acond}) essentially means that $A$ consists of
$(KL)$
lowering operators.
Let $\{\sigma_i\}_{i=1,\ldots,K+L}$ be such that $\sigma_i=\pm1$
and
$\sum_{i=1}^{K+L}\sigma_i=KL$.
Then for any integer $k$ with $0\le k\le K+L$, we have
\begin{eqnarray}
&&\abs{
\gs{\rbk{\prod_{i=1}^k\osig{i}}A\rbk{\prod_{i=k+1}^{K+L}\osig{i}}}
\frac{2^{2J}(J!)^2}{(2J)!}\gs{\kone^J\cbk{A\kplus^{(KL)}}\kone^J}
}
\ret
&&\le\delta(A;K,L),
\label{dBound1}
\end{eqnarray}
where $J=\min\{K,L\}$, and
\begin{equation}
\abs{
\gs{\rbk{\prod_{i=1}^k\osig{i}}A\rbk{\prod_{i=k+1}^{K+L}\osig{i}}}
}
\le 3\delta(A;K,L).
\label{dBound2}
\end{equation}
Here $\delta(A;K,L)$ is given by
\begin{equation}
\delta(A;K,L):=\frac{1}{2}(aN)(2oN)^{KL}b_J,
\label{delta1}
\end{equation}
for general $A$.
For $A=1$, in which case only $K=L$ is allowed, we can set
\begin{equation}
\delta(1;K,K)=\frac{1}{2}b_K.
\label{delta2}
\end{equation}
\label{hardLemma}
\end{lemma}
The lemma will be proved after completing the proof of the main
theorem.
We again want to control the quantity $\Delta^{(M)}$ in
(\ref{DeltaDef}).
By using the relation $U\osig{}U^{1}=O^{\sigma}$ (which
follows form
(\ref{commutation3})), we find that $\Delta^{(M)}$ can be written in
terms of
a double commutator as follows.
\begin{eqnarray}
&&
2N\norm{\Psi^{(M)}}^2\Delta^{(M)}
\ret
&=& 2\gs{\kminus^MH\kplus^M}2E\gs{\kminus^M\kplus^M}
\ret
&=& \gs{\kminus^MH\kplus^M}+\gs{\kplus^MH\kminus^M}
\ret
&& \gs{\kminus^M\kplus^MH}\gs{H\kplus^M\kminus^M}
\ret
&=& \gs{[\kminus^M,[H,\kplus^M]]}
\ret
&=& \sum_{m=0}^{M1} \gs{[\kminus^M,\kplus^m[H,\Oplus]
\kplus^{Mm1}]}
\ret
&=&
\sum_{m=1}^{M1}\sum_{\ell=0}^{M1}\sum_{n=0}^{m1}
\ret
&& \gs{\kminus^\ell\kplus^n[\Ominus,\Oplus]\kplus^{mn
1}[H,\Oplus]
\kplus^{Mm1}\kminus^{M\ell1}}
\ret
&+& \sum_{m=0}^{M1}\sum_{\ell=0}^{M1}
\ret
&&\gs{\kminus^\ell\kplus^m[\Ominus,[H,\Oplus]]
\kplus^{Mm1}\kminus^{M\ell1}}
\ret
&+&
\sum_{m=0}^{M2}\sum_{\ell=0}^{M1}\sum_{n=0}^{Mm2}
\ret
&&
\gs{\kminus^\ell\kplus^m[H,\Oplus]\kplus^n[\Ominus,\Oplus]
\kplus^{Mmn2}\kminus^{M\ell1}}.
\end{eqnarray}
Note that the symmetry vi), along with the relation $UCU^{1}=C$
(which
follows from (\ref{commutation1}) and (\ref{commutation3})),
implies that $C\Phi=0$.
By also using the relation $[\Ominus,\Oplus]=2\gamma C$,
and the
fact that $O^\pm$ are the raising and the lowering operators,
we can
bound the above quantity as
\begin{eqnarray}
&&\abs{2N\norm{\Psi^{(M)}}^2\Delta^{(M)}}
\ret
&\le& \sum_{m=1}^{M1}\sum_{\ell=0}^{M1}\sum_{n=0}^{m1}
\ret
&& 2\gamma M
\abs{\gs{\kminus^\ell\kplus^{m1}[H,\Oplus]
\kplus^{Mm1}\kminus^{M\ell1}}}
\ret
&+& \sum_{m=0}^{M1}\sum_{\ell=0}^{M1}
\ret
&&\abs{\gs{\kminus^\ell\kplus^m[\Ominus,[H,\Oplus]]
\kplus^{Mm1}\kminus^{M\ell1}}}
\ret
&+&
\sum_{m=0}^{M2}\sum_{\ell=0}^{M1}\sum_{n=0}^{Mm2}
\ret
&&
2\gamma M
\abs{\gs{\kminus^\ell\kplus^m[H,\Oplus]
\kminus^{Mm2}\kminus^{M\ell1}}}
\ret
&\le&
6\gamma M^4\delta([H,\Oplus];M2,M1)
+3M^2\delta([\Ominus,[H,\Oplus]];M1,M1),
\end{eqnarray}
where we have used (\ref{dBound2}).
The use of Lemma~\ref{hardLemma} is justified since $M$ satisfies
(\ref{M/N<}) which (with $K+L\le 2M$) guarantees the condition
(\ref{K+Lcond}).
By using (\ref{delta1}) and then the bound (\ref{b/b}), we get
\begin{eqnarray}
&&\abs{2N\norm{\Psi^{(M)}}^2\Delta^{(M)}}
\ret
&\le& 3\gamma M^4(4rohN)(2oN)b_{M2}
+\frac{3}{2}M^2(16r^2o^2hN)b_{M1}
\ret
&\le&\cbk{\frac{24\gamma rh}{o^2\mu^4}\frac{M^4}{N^2}
+\frac{24r^2h}{\mu^2}\frac{M^2}{N}}b_M.
\label{Bunshi}
\end{eqnarray}
On the other hand, from (\ref{dBound1}) and (\ref{delta2}), we find
\begin{equation}
\norm{\Psi^{(M)}}^2=\gs{\kminus^M\kplus^M}\ge\frac{1}{2}b_M.
\label{Bunbo}
\end{equation}
By combining (\ref{Bunshi}) and (\ref{Bunbo}), and substituting the
assumed
bound (\ref{M/N<}), we finally get
\begin{eqnarray}
\abs{\Delta^{(M)}}
&\le&
\frac{24r^2h}{\mu^2}\cbk{1+\frac{\gamma}{o^2\mu^2r}\frac{M^2}{N}}
\rbk{\frac{M}{N}}^2
\ret
&\le& c_3\rbk{\frac{M}{N}}^2,
\end{eqnarray}
with $c_3=24r^2h\mu^{2}\cbk{1+\gamma o^{2}\mu^{2}r^{1}c_2}$.
It remains to prove Lemma~\ref{hardLemma}.
We prepare the following.
\begin{lemma}
Let $K,L$ be nonnegative integers which satisfy (\ref{K+Lcond}).
We assume that the operator $A$ satisfies the conditions of
Lemma~\ref{hardLemma}.
Let $\{\sigma_i\}_{i=1,\ldots,K+L}$, $\{\tau\}_{i=1,\ldots,K+L}$
be such that
$\sigma_i=\pm1$, $\tau_i=\pm1$, and
$\sum_{i=1}^{K+L}\sigma_i=\sum_{i=1}^{K+L}\tau_i=KL$.
Then for any integers $k,\ell$ with $0\le k,\ell\le K+L$, we have
\begin{equation}
\abs{
\gs{\rbk{\prod_{i=1}^k\osig{i}}A\rbk{\prod_{i=k+1}^{K+L}\osig{i}}}

\gs{\rbk{\prod_{i=1}^\ell
O^{\tau_i}}A\rbk{\prod_{i=\ell+1}^{K+L}O^{\tau_i}}}
}
\le
\delta(A;K,L),
\label{dBound3}
\end{equation}
with the same $\delta(A;K,L)$ as in (\ref{delta1}) and (\ref{delta2}).
\label{subLemma}
\end{lemma}
\begin{proof}{Proof of Lemma~\ref{hardLemma} given
Lemma~\ref{subLemma}}
Let $B$ be an arbitrary operator which satisfies $[B,C]=0$.
Then
\begin{eqnarray}
\gs{\kone^JB\kone^J}
&=&
\gs{\rbk{\frac{\Oplus+\Ominus}{2}}^JB\rbk{\frac{\Oplus+\Ominus}{2
}}^J}
\ret
&=& \sum_{\tau_1=\pm1}\cdots\sum_{\tau_{2J}=\pm1}
2^{
2J}\gs{\rbk{\prod_{i=1}^JO^{\tau_i}}B\rbk{\prod_{i=J+1}^{2J}O^{\tau
_i}}}
\ret
&=&
\sum_{\{\tau_i\};\sum\tau_i=0}
2^{
2J}\gs{\rbk{\prod_{i=1}^JO^{\tau_i}}B\rbk{\prod_{i=J+1}^{2J}O^{\tau
_i}}},
\label{OBO1}
\end{eqnarray}
where the final sum is over all $\tau_i=\pm1$ with
$\sum_{i=1}^{2J}\tau_i=0$.
The constraint comes from the fact that $\Phi$ is an eigenstate of
the $U(1)$
generator $C$.
Since the number of distinct combinations $\{\tau_i\}$ which
satisfy the
constraint is equal to $\JJ=(2J)!/(J!)^2$, we can rewrite
(\ref{OBO1}) as
\begin{equation}
\JJ^{1}\sum_{\{\tau_i\};\sum\tau_i=0}
\gs{\rbk{\prod_{i=1}^JO^{\tau_i}}B\rbk{\prod_{i=J+1}^{2J}O^{\tau_i}}
}
=
\frac{2^{2J}(J!)^2}{(2J)!}\gs{\kone^JB\kone^J}
\label{OBO2}
\end{equation}
To prove (\ref{dBound1}), we set $B=A\kplus^{KL}$, and substitute
(\ref{OBO2}) into the lefthand side of (\ref{dBound1}) to get
\begin{eqnarray}
&&
\abs{
\gs{\rbk{\prod_{i=1}^k\osig{i}}A\rbk{\prod_{i=k+1}^{K+L}\osig{i}}}
\frac{2^{2J}(J!)^2}{(2J)!}\gs{\kone^JA\kplus^{KL}\kone^J}
}
\ret &\le&
\JJ^{1}\sum_{\{\tau_i\};\sum\tau_i=0}
\ret &&
\abs{
\gs{\rbk{\prod_{i=1}^k\osig{i}}A\rbk{\prod_{i=k+1}^{K+L}\osig{i}}}
 \gs{\rbk{\prod_{i=1}^JO^{\tau_i}}A\kplus^{KL}
\rbk{\prod_{i=J+1}^{2J}O^{\tau_i}}}
}
\ret &\le&
\JJ^{1}\sum_{\{\tau_i\};\sum\tau_i=0}
\delta(A;K,L)
\ret
&=&\delta(A;K,L),
\end{eqnarray}
where we have used (\ref{dBound3}).
To prove (\ref{dBound2}), we simply substitute the estimate
\begin{eqnarray}
&&\abs{
\frac{2^{2J}(J!)^2}{(2J)!}
\gs{\kone^JA\kplus^{KL}\kone^J}
}
\ret
&& \le \norm{A\kplus^{KL}}\frac{2^{2J}(J!)^2}{(2J)!}
\gs{\kone^{2J}}
\le
2\delta(A;K,L),
\end{eqnarray}
which follows from the definition of norm (\ref{DEFnorm}), into the bound
(\ref{dBound1}).%
\end{proof}
\begin{proof}{Proof of Lemma~\ref{subLemma}}
The desired bound (\ref{dBound3}) is trivial if $K=L=0$.
We shall prove the bound inductively in $K+L$.
Fix $K,L$ with $K\ge L$ (the bounds for $KL$}\\
\frac{3}{2}(4raoN)(2oN)b_{L1}
&\mbox{if $K=L$}
\end{array}\right.
\ret &\le&
\frac{3}{2}(4raoN)(2oN)^{KL+1}b_{L1},
\end{eqnarray}
for $K\ge L$, where we used the bound (\ref{b/b}) in the case $K>L$.
When $\sigma_j=1$, we also get
\begin{eqnarray}
&&
\abs{\gs{\rbk{\prod_{i=1}^{j1}\osig{i}}[\Ominus,A]
\rbk{\prod_{i=j+1}^{K+L}\osig{i}}}}
\ret &\le&
3 \delta([\Ominus,A];K,L1)
\ret &\le&
\frac{3}{2}(4raoN)(2oN)^{KL+1}b_{L1}.
\end{eqnarray}
Since there are at most $(K+L)$ terms in the sum in (\ref{D1<}), we
find that
\begin{equation}
D_1\le(K+L)\frac{3}{2}(4raoN)(2oN)^{KL+1}b_{L1}.
\label{D1<2}
\end{equation}
It is obvious that the quantity $D_3$ satisfies the same bound
as (\ref{D1<2}).
To evaluate the quantity $D_2$ in (\ref{Ddecomp}), we transform
$\{\sigma_i\}_{i=1,\ldots,K+L}$ into $\{\tau_i\}_{i=1,\ldots,K+L}$
by
successively exchanging neighboring indices.
We then get
\begin{equation}
D_2\le\sum_{\{\kappa_i\}}
\abs{\gs{A\rbk{\prod_{i=1}^{j1}O^{\kappa_i}}
[O^{\kappa_j},O^{\kappa_{j+1}}]\rbk{\prod_{i=j+2}^{K+L}O^{\kappa_i}}
}},
\end{equation}
where $\{\kappa_i\}_{i=1,\ldots,K+L}$ is summed over the
sequence of
configurations which interpolates between
$\{\sigma_i\}_{i=1,\ldots,K+L}$
and $\{\tau_i\}_{i=1,\ldots,K+L}$, and $j$ (which depends on
$\{\kappa_i\}_{i=1,\ldots,K+L}$) indicates where the indices are
exchanged.
By using the commutation relation (\ref{commutation3}), and
$C\Phi=0$, we
can further bound $D_2$ as
\begin{eqnarray}
D_2 &\le&
\sum_{\{\kappa_i\}}
2\gamma(K+L)\abs{\gs{A\rbk{\prod_{i=1}^{j1}O^{\kappa_i}}
\rbk{\prod_{i=j+2}^{K+L}O^{\kappa_i}}}}
\ret
&\le&
\sum_{\{\kappa_i\}}
6\gamma(K+L)\delta(A;K1,L1)
\ret
&\le&
3\gamma KL(K+L)(aN)(2oN)^{KL}b_{L1},
\label{D2<}
\end{eqnarray}
where we have used the fact that at most $KL$ exchanges are
necessary to get
$\{\tau\}_{i=1,\ldots,K+L}$ from $\{\sigma_i\}_{i=1,\ldots,K+L}$,
and the
bound (\ref{dBound2}).
By substituting the bounds (\ref{D1<2}) and (\ref{D2<}) into the
decomposition
(\ref{Ddecomp}), and using the bounds (\ref{b/b}) and (\ref{delta1}),
we
finally get
\begin{eqnarray}
D &\le&
\cbk{24(K+L)ro^2N+3\gamma KL(K+L)}(aN)(2oN)^{KL}b_{L1}
\ret
&\le&
\cbk{\frac{48r}{\mu^2}\frac{K+L}{N}
+\frac{6\gamma}{o^2\mu^2}\frac{KL(K+L)}{N^2}}
\delta(A;K,L)
\ret
&\le& \delta(A;K,L),
\end{eqnarray}
where we used the assumption (\ref{K+Lcond}) and $KL\le(K+L)^2/4$.
This proves the desired (\ref{dBound3}) for $A\ne1$.
The case $A=1$ is much easier.
One notes that only $D_2$ is nonvanishing in the decomposition
(\ref{Ddecomp}).
The similar estimate as the above proves the desired result.%
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\appendix
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Ground states of infinite systems}
\label{APgs}
In the present Appendix, we give mathematically precise definitions of
ground states in an infinite system, and discuss relations between
different definitions.
The contents of the present Appendix might be wellknown to
experts, but they have not been written down
explicitly as far as we know.
We think it would be convenient for the readers to have
them included in the present paper.
We start by briefly reviewing basic setups in operator algebraic
approach to quantum systems with infinitely many degrees of
freedom \cite{BratteliRobinson,Ruelle}.
For simplicity we shall consider a quantum manybody system
defined on the $d$dimensional hypercubic lattice $\Zd$.
With each site $x\in\Zd$, we associate a finite dimensional Hilbert
space $\calH_x$ which is assumed to be identical to $\calH_o$
where
$o$ is a fixed site (the origin) of $\Zd$.
The Hilbert space corresponding to a finite subset
$\Omega\subset\Zd$ is
\begin{equation}
\calH_\Omega:=\bigotimes_{x\in\Omega}\calH_x.
\label{Hilb2}
\end{equation}
Let $\calA_\Omega$ denote the set of all the operators on
$\calH_\Omega$.
The basic object in the operator algebraic approach is the algebra of
quasi local operators defined as
\begin{equation}
\calA:=\overline{\bigcup_{\Omega}\calA_\Omega},
\label{DEFalgebra}
\end{equation}
where the union is over all the finite subsets $\Omega\subset\Zd$,
and
the completion is taken with respect to the norm (\ref{DEFnorm})
for local operators.
Note that we have made $\calA$ into a Banach space.
A state $\rho(\cdots)$ is a linear map from $\calA$ to ${\bf C}$
which
satisfies $\rho(1)=1$, and $\rho(A^*A)\ge0$
for any $A\in\calA$.
It can be shown \cite{BratteliRobinson} that it automatically holds
that $\abs{\rho(A)}\le\norm{A}$, and $\rho(A^*)=\rho(A)^*$.
We denote by $\calE$ the set of all states on $\calA$.
Since $\calE$ is the unit sphere of the dual space $\calA^*$, the
BanachAlaoglu theorem \cite{ReedSimon} implies that $\calE$ is
compact in the weak$*$ topology.
The compactness provides us with a useful way of constructing
states on $\calA$.
Let $\{\Omega_i\}_{i=1,2,\ldots}$ be an arbitrary sequence of finite
subsets of $\Zd$ which tends to $\Zd$ in the sense of van Hove
\cite{VanHove,Ruelle} as
$i\toinf$.
For each $i$ we take a state (density matrix) $\rho_i(\cdots)$ on
the algebra $\calA_{\Omega_i}$.
Since $\rho_i(\cdots)$ can be naturally regarded as an element of
$\calE$ (by defining, for example, $\rho(A)=0$ for
$A\in\calA\backslash\calA_{\Omega_i}$), the compactness ensures
that one can take a subsequence
$\{i(j)\}_{j=1,2,\ldots}\subset\{1,2,\ldots\}$ such that the
weak$*$ limit
\begin{equation}
\rho(\cdots):=\lim_{j\toinf}\rho_{i(j)}(\cdots)
\label{weak*limit}
\end{equation}
exists.
In the physicists' language, (\ref{weak*limit}) should be read
\begin{equation}
\rho(A)=\lim_{j\toinf}\rho_{i(j)}(A),
\end{equation}
for each $A\in\calA$.
As in Section~2, we let $h_o$ be the local Hamiltonian at the origin
$o\in\Zd$, which acts on the finite dimensional Hilbert space
$\bigotimes_{x\in\calS_o}\calH_x$ with the support set $\calS_o$
containing $r$ sites.
We also set $h_x=\tau_x(h_o)$, and, for any finite set
$\Omega\in\Zd$,
\begin{equation}
H_{\Omega}:=\sum_{x\in\Omega}h_x,
\end{equation}
where $\tau_x$ denotes the translation by the lattice vector $x$.
We now describe three different definitions of the set of {\em
ground states}.
The first definition is standard in mathematical literature, and is
\begin{equation}
\calG_1:=\{\omega\in\calE \vbar
\omega(A^*[H_{\barOmega},A])\ge0
\quad\mbox{for any $A\in\calA_\Omega$, and for any finite
$\Omega\subset\Zd$}\}.
\end{equation}
Here we introduced
\begin{equation}
\barOmega:=\{x \vbar \calS_x\cap\Omega\ne\emptyset\},
\end{equation}
where $\calS_x=\tau_x(\calS_o)$ is the support set for $h_x$.
(We use the same symbol $\tau_x$ to denote
the translation operators for
subsets of $\Zd$ and that for operators.)
The second definition is due to Aizenman and Lieb
\cite{AizenmanLieb}.
(See also \cite{AizenmanDaviesLieb}.)
The definition is useful because of its similarity to ``classical''
definitions of ground states.
It is
\begin{equation}
\calG_2:=\{\omega\in\calE \vbar
\omega(H_{\barOmega})\le\omega(T(H_{\barOmega}))
\quad\mbox{for any $T\in\calP_\Omega$, and for any finite
$\Omega\subset\Zd$}\},
\end{equation}
where $\calP_\Omega$ is the set of all local perturbations on
$\Omega$.
A local perturbation $T$ on $\Omega$ is a linear mapping
$T:\calA\to\calA$ which satisfies $T(A)\ge0$ for any $A\ge0$, and
$T(A)=A$ for any $A\in\calA_{\Omega^{\rm c}}$, where
$\calA_{\Omega^{\rm
c}}:=\overline{\bigcup_{\Omega'}\calA_{\Omega'\backslash\Omega}}
$
is the operator algebra outside of $\Omega$.
The third definition already appeared in Sections~1.2 and 2.5,
and is probably the simplest among the three definitions.
(Essentially the same definition can be found in
\cite{AffleckKennedy}.)
It is
\begin{equation}
\calG_3:=\{\omega\in\calE \vbar \omega(h_x)=\epsilon_0
\quad\mbox{for any $x\in\Zd$}\},
\end{equation}
where the ground state energy density $\epsilon_0$ is defined as
follows.
Let $\Lambda$ be the $d$dimensional $L\times\cdots\times L$
hypercubic lattice.
We define the corresponding Hamiltonian with periodic boundary
conditions as
\begin{equation}
\hampbc=\sum_{x\in\Lambda}h_x,
\label{Hampbc}
\end{equation}
where, for a site $x$ close to the boundary of $\Lambda$, we
identify $h_x$
in (\ref{Hampbc}) as an operator in $\calA_\Lambda$ by imposing
periodic boundary conditions.
Note that a Hamiltonian with periodic boundary conditions is
simply denoted as $\ham$ except in the present Appendix.
Then we define $\epsilon_0$ by
\begin{equation}
\epsilon_0:=\lim_\TDL\inf_{\rho_\Lambda\in\calA_\Lambda}
\frac{1}{\Lambda}\rho_\Lambda(\hampbc),
\label{DEFe0}
\end{equation}
where $\Lambda$ is the number of sites in $\Lambda$, and
the existence of the limit can be proved by a standard
argument.
For each $i=1,2,3$, we denote by $\hat{\calG}_i$ the set of
$\omega\in\calG_i$ which is translation invariant, {\em i.e.\/},
$\omega(\tau_x(A))=\omega(A)$ for any $A\in\calA$ and any
$x\in\Zd$.
Now we discuss the relations between these different definitions.
We first note the following.
\begin{pro}
We have $\calG_1=\calG_2$.
\label{G1=G2}
\end{pro}
\begin{proof}{Outline of proof}
Nontrivial parts of the proof is worked out in literature, and we only
have to make some formal observations.
We make use of the results summarized as Theorem~6.2.52 in
\cite{BratteliRobinson}.
To prove $\calG_1\subset\calG_2$, we note that the above
mentioned theorem in \cite{BratteliRobinson} says that
$\omega\in\calG_1$ if and only if
\begin{equation}
\omega(H_{\barOmega})\le\omega'(H_{\barOmega}),
\end{equation}
for any $\omega'\in\calE$ such that $\omega(B)=\omega'(B)$ for all
$B\in\calA_{\Omega^{\rm c}}$, and for any finite $\Omega\in\Zd$.
By choosing the perturbed state $\omega'(\cdots)$ in a special form
$\omega(T(\cdots))$ as in the definition of $\calG_2$, we see that
$\omega\in\calG_2$.
To prove $\calG_2\subset\calG_1$, we can follow the part (1)
$\Rightarrow$ (2) of the proof of the above mentioned theorem in
\cite{BratteliRobinson} without any modifications.%
\end{proof}
Next we note that
\begin{pro}
We have $\calG_1=\calG_2\supset\calG_3$.
\label{G3}
\end{pro}
\begin{proof}{Proof}
Because of Proposition~\ref{G1=G2}, it suffices to show
$\calG_3\subset\calG_2$.
The proof is elementary.
We want to get a contradiction out of the assumption that there is a
state $\omega$ such that $\omega\in\calG_3$ and
$\omega\not\in\calG_2$.
>From the assumption there exists a finite set $\Omega\subset\Zd$,
a local perturbation $T\in\calP_\Omega$, and a constant
$\varepsilon>0$ such that
\begin{equation}
\omega(H_{\barOmega})\omega(T(H_{\barOmega}))\ge\varepsilon.
\label{omegaT}
\end{equation}
Let $\ell$ be an integer such that $\barOmega$ is contained in a
suitable $d$dimensional $\ell\times\cdots\times\ell$ hypercubic
lattice $\Lambda_0$.
For an integer $n$, let $\Lambda$ be the $d$dimensional
$(n\ell)\times\cdots\times(n\ell)$ hypercubic lattice.
There are translation operators $\tau_i$ with $i=1,2,\ldots,n^d$
such that
\begin{equation}
\Lambda=\bigcup_{i=1}^{n^d}\tau_i(\Lambda_0).
\end{equation}
Let $\omega_\Lambda$ be the state obtained by simply restricting
$\omega$ onto $\calA_\Lambda$.
We further define
\begin{equation}
\omega'_\Lambda(A):=\omega_\Lambda(\tilde{T}(A)),
\end{equation}
for a suitable $A\in\calA_\Lambda$, where
\begin{equation}
\tilde{T}=\prod_{i=1}^{n^d}{\tau_i}^{1}T\tau_i.
\end{equation}
By using (\ref{omegaT}) and the properties of local perturbations,
we
observe that
\begin{equation}
\omega_\Lambda(\hampbc)\omega'_\Lambda(\hampbc)
=\sum_{i=1}^{n^d}\cbk{\omega_\Lambda(H_{\tau_i(\barOmega)})
\omega_\Lambda(T(H_{\tau_i(\barOmega)}))}
\ge n^d\varepsilon.
\label{bulk}
\end{equation}
On the other hand, since we have $\omega(h_x)=\epsilon_0$, we get
\begin{equation}
\omega_\Lambda(\hampbc)\le
(n\ell)^d\epsilon_0+\beta(n\ell)^{d1},
\label{surface}
\end{equation}
where $\beta$ is a finite constant which takes care of the boundary
effects.
By combining the bounds (\ref{bulk}) and (\ref{surface}), we get
\begin{equation}
(n\ell)^{d}\omega'_\Lambda(\hampbc)\epsilon_0
\le \ell^{d}\varepsilon+\beta(n\ell)^{1},
\end{equation}
which contradicts with the definition (\ref{DEFe0})
of $\epsilon_0$ by taking $n$
sufficiently large.%
\end{proof}
It should be noted that we do not have $\calG_1=\calG_2=\calG_3$.
For example, a state with a single domain wall
in the Ising model belongs to
$\calG_1=\calG_2$ but not to $\calG_3$.
It is a delicate problem to decide which definition is more
``realistic''.
As for the translation invariant ground states, however, we have the
following rather satisfactory result.
\begin{pro}
We have $\hat{\calG}_1=\hat{\calG}_2=\hat{\calG}_3$.
\end{pro}
\begin{proof}{Proof}
Because of Propositions~\ref{G1=G2} and \ref{G3}, it suffices to
show that $\hat{\calG}_1\subset\calG_3$.
Again the most essential part can be found in literature.
In Theorem~6.2.58 of \cite{BratteliRobinson}, it is proved that a
translation invariant state $\omega$ belongs to
$\hat{\calG}_1$ if and
only
if $\omega(h_x)=\tilde{\epsilon}_0$ for any $x\in\Zd$.
The ground state energy density is defined as
\begin{equation}
\tilde{\epsilon}_0:=\inf_{\omega'\in\calE_{\rm inv}}\omega'(h_x),
\end{equation}
where $\calE_{\rm inv}$ is the set of translation invariant states in
$\calE$.
We only have to show that $\epsilon_0=\tilde{\epsilon}_0$,
and this may be done in several ways.
Here we offer a simple
constructive proof.
For each $\Lambda$, we can take a ground state
$\Phi_\Lambda\in\calH_\Lambda$ of the Hamiltonian $\hampbc$,
which state is invariant under translations that take into account
the periodic boundary conditions imposed on $\Lambda$.
Define a state $\omega\in\calE$ by the (weak$*$) limiting
procedure (\ref{infGSseq}).
By construction, we see that $\omega\in\hat{\calG}_1$.
The above mentioned theorem of \cite{BratteliRobinson} then
implies that $\omega(h_x)=\tilde{\epsilon}_0$.
On the other hand, our definition (\ref{DEFe0})
of $\epsilon_0$ implies that
$\omega(h_x)=\epsilon_0$.%
\end{proof}
One might be interested to know if there is any general theorem
which tells us exactly what are the elements of the above sets of
ground
states.
The following is an example of such general theorems.
It establishes uniqueness of the ground state when there
is a decoupled Hamiltonian with a unique
ground state, and then one adds a weak (but completely
arbitrary) translation invariant perturbation to the model.
Suppose that the Hamiltonian at the origin can be written as
\begin{equation}
h_o=v_o+\delta p_o.
\end{equation}
The main part $v_o$ acts only on the space $\calH_o$, and its
lowest eigenvalue is simple.
The perturbation $p_o$ is an arbitrary selfadjoint operator on
$\bigotimes_{x\in\calS_o}\calH_x$, and $\delta$ is a constant.
By using a rigorous perturbation technique, the following was proved
in \cite{KennedyTasaki}.
\begin{theorem}
There exists a finite constant $\delta_0>0$ which depends on the
dimension $d$, and on the operators $v_o$ and $p_o$.
For $\delta\le\delta_0$, the set of ground states
$\hat{\calG}_1=\hat{\calG}_2=\hat{\calG}_3$ consists of a unique
element.
\label{UniqueGS}
\end{theorem}
For example the Ising model under sufficiently large transverse
magnetic field (\ref{Ising1}) is covered by the above theorem by
setting $h_o=S^{(1)}_o$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Pure infinite volume ground state in systems with
discrete symmetry breaking}
\label{APpure}
In the present Appendix, we concentrate on a system in which
a discrete symmetry is spontaneously broken.
We assume, for each finite system,
the existence of an ``obscured
symmetry breaking'' and the existence of an energy gap above the
first lowlying eigenstate.
Then we can prove that, by forming a linear combination of the
(finite volume) ground state and the lowlying state, and then
taking an infinite volume limit, one indeed gets a
{\em pure\/} infinite volume ground state.
As far as we know this is the first rigorous and general
result which explicitly
tells one how to construct a pure infinite volume ground state when
there is a symmetry breaking.
The theorem is desirable in this sense, but we have to note that
the assumption on the existence of a gap is a rather strong one,
which is not at all easy to verify even
in relatively simple problems\footnote{
Even in the models (like the transverse Ising model of Section~1.2)
where one has a convergent cluster expansion, it may not be
easy to verify the existence of the gap.
As for the transverse Ising model in one dimension, one
can make use of the mapping to the free fermion problem to
control the gap.
}.
We also stress that the techniques involved here crucially depend
on the fact that there is only one lowlying
eigenstate.
To prove
the corresponding conjecture (stated in
Section~2.5) for the models with broken continuous symmetry
seems formidably difficult at present.
We study the situation basically identical to that in
Section~2.1, but with additional assumption on the translation
invariance.
The translation invariance is by no means essential in
proving the main theorem, but the implication of the theorem
is interesting only in translation invariant systems.
Let $\Lambda$ be a $d$dimensional
hypercubic lattice with periodic boundary conditions, and
denote by $N$ the number of sites in $\Lambda$.
We consider a quantum manybody system on $\Lambda$ as in
Section~2.1.
The Hilbert space is constructed as in (\ref{Hilb}), the
Hamiltonian as (\ref{ham}), and the order operator as
(\ref{ord}).
The additional assumptions are that we have
$h_x=\tau_x(h_o)$ and $o_x=\tau_x(o_o)$, where $\tau_x$ is the
translation operator that takes into account the periodic
boundary conditions.
We also require that each $o_x$ acts only on the local Hilbert
space $\calH_x$.
In some situations, one might need to redefine the notion of
``sites'' to satisfy the translation invariance.
See Section~3.2.
Let $E_\Lambda^{(0)}$, $E_\Lambda^{(1)}$
with $E_\Lambda^{(0)}0$.
We also assume that the ground state $\PLO$ exhibits an
``obscured symmetry breaking'' as
\begin{equation}
(\PLO,\op\,\PLO)=0,
\label{Ohat1}
\end{equation}
\begin{equation}
(\PLO,(\op)^2\,\PLO)\ge(\mu oN)^2,
\label{LROap}
\end{equation}
with a constant $\mu>0$, and
\begin{equation}
(\PLO,(\op)^3\,\PLO)=0.
\label{Ohat3}
\end{equation}
Although we did not assume the condition (\ref{Ohat3}) in Section~1.2,
it is valid in most situations.
We shall again consider the lowlying state of Horsch and von der
Linden \cite{HorschLinden}
\begin{equation}
\Psi_\Lambda:=\frac{\op\PLO}{\norm{\op\PLO}},
\label{Psiap}
\end{equation}
and its linear combination with the ground state
\begin{equation}
\XL:=\frac{1}{\sqrt{2}}\rbk{\PLO+\Psi_\Lambda},
\label{XiAp}
\end{equation}
which was first considered by Kaplan, Horsch, and von der Linden
\cite{KaplanHorschLinden}.
>From a straightforward calculation using the
definitions (\ref{Psiap}), (\ref{XiAp}), and the
assumed (\ref{Ohat1}), (\ref{LROap}), and (\ref{Ohat3}), we find
\cite{KaplanHorschLinden} that
the above state (\ref{XiAp}) exhibits a symmetry breaking as
\begin{eqnarray}
(\XL,\op\,\XL)&=&\frac{1}{2}
\cbk{(\PLO,\op\,\PLO)
+\frac{2(\PLO,(\op)^2\,\PLO)}{\norm{\op\PLO}}
+\frac{(\PLO,(\op)^3\,\PLO)}{\norm{\op\PLO}^2}
}
\ret
&=&\sqrt{(\PLO,(\op)^2\,\PLO)}\ge \mu o N.
\label{OrderAP}
\end{eqnarray}
Let $A$ be a local selfadjoint operator which acts on
$\bigotimes_{x\in\calS'}\calH_x$, where the
number of sites in the support set $\calS'$ is bounded by
a constant $r'$.
For a subset $\Omega\subset\Lambda$, we set
\begin{equation}
\AO:=\sum_{x\in\Omega}\tau_x(A),
\end{equation}
where $\tau_x(A)$ is a translate of $A$ by a lattice vector $x$.
Then the main result of the present Appendix is the following.
\begin{theorem}
We have
\begin{equation}
\lim_{\Omega\toinf}\lim_{N\toinf}
\frac{1}{\Omega^2}\cbk{(\XL,\AOS\,\XL)(\XL,\AO\,\XL)^2}
=0,
\label{nofluc}
\end{equation}
for any local operator $A$.
\label{pureth}
\end{theorem}
>From the Definition~\ref{DEFpure} of pure state, we get the
following interesting conclusion.
\begin{coro}
The infinite volume ground state
\begin{equation}
\omega_+(\cdots)=\lim_{N\toinf}(\XL,(\cdots)\,\XL),
\end{equation}
defined by taking a suitable subsequence, is a pure translation invariant
ground state.
\label{purecoro}
\end{coro}
It is obvious that the same is true for the infinite volume ground state
$\omega_(\cdots)$ constructed from $(\PLO\PsL)/\sqrt{2}$
instead of (\ref{XiAp}).
In the following we prove Theorem~\ref{pureth}.
For simplicity, we drop the subscript $\Lambda$ from
$\PLO$, $\PL^{(1)}$, $\PsL$, $\ham$, $\op$, etc.
We start from the following lemma which provides us with the basic
tool in the proof.
In short the lemma says that the set of two states $\{\Phi^{(0)},
\Phi^{(1)}\}$ can be used almost as a ``complete basis'' in some situations.
\begin{lemma}
Let $B$ and $C$ be arbitrary selfadjoint operators.
Then for $i,j=0$ or $1$, we have
\begin{eqnarray}
&&\abs{\amp{i}{BC}{j}
\sum_{k=0,1}\amp{i}{B}{k}\amp{k}{C}{j}}
\ret
&=&\abs{\amp{i}{B\calP C}{j}}
\ret
&\le&\frac{\sqrt{\norm{[B,[H,B]]}\norm{[C,[H,C]]}}}{2\EG},
\label{EGbound}
\end{eqnarray}
where $\calP$ is the projection operator onto the space orthogonal
to both $\Phi^{(0)}$ and $\Phi^{(1)}$.
\end{lemma}
\begin{proof}{Proof}
>From the existence of a gap as in (\ref{gapCond}), we get
the operator inequality
$\calP\le(HE_i)/\EG$ for $i=0,1$.
By using the Schwartz inequality, we have
\begin{eqnarray}
&&\abs{\amp{i}{B\calP C}{j}}^2
\ret
&\le&\amp{i}{B\calP B}{i}\amp{j}{C\calP C}{j}
\ret
&\le&\amp{i}{B\frac{HE_i}{\EG}B}{i}
\amp{j}{C\frac{HE_i}{\EG}C}{j}
\ret
&=&(2\EG)^{2}\amp{i}{[B,[H,B]]}{i}\amp{j}{[C,[H,C]]}{j}
\ret
&\le&(2\EG)^{2}\norm{[B,[H,B]]}\norm{[C,[H,C]]},
\end{eqnarray}
which is the desired bound.%
\end{proof}
As the first application of the lemma, we state the following result
which is both useful and important.
The lemma says that the lowlying state (\ref{Psiap}) is indeed a very good
approximation of the first excited state $\Phi^{(1)}$.
\begin{lemma}
One can redefine the (quantum mechanical)
phase of the first excited state $\Phi^{(1)}$
so that the bound
\begin{equation}
\norm{\Phi^{(1)}\Psi}^2\le\frac{4hr^2}{\EG\mu^2}\frac{1}{N}
\label{1=PsiA}
\end{equation}
holds.
\label{1=PsiLemma}
\end{lemma}
\begin{proof}{Proof}
Since $(\Phi^{(0)},\Psi)=0$, we can write $\Psi=\alpha\Phi^{(1)}+\Psi'$,
where $\Psi'=\calP\Psi$.
By redefining the phase of $\Phi^{(1)}$, we can choose $\alpha\ge0$.
First note that
\begin{eqnarray}
\norm{\Phi^{(1)}\Psi}^2&=&
\norm{(1\alpha)\Phi^{(1)}\Psi'}^2
\ret
&=&(1\alpha)^2+\norm{\Psi'}^2\le2\norm{\Psi'}^2,
\end{eqnarray}
where we have used
$(1\alpha)^2\le1\alpha\le1\alpha^2=\norm{\Psi'}^2$.
To bound $\norm{\Psi'}$, we use (\ref{EGbound}) to get
\begin{eqnarray}
\norm{\Psi'}^2&=&(\Psi,\calP\,\Psi)
=\frac{\amp{0}{O\calP O}{0}}{\amp{0}{O^2}{0}}
\ret
&\le&\frac{\norm{[O[H,O]]}}{2\EG\amp{0}{O^2}{0}}
\ret
&\le&\frac{4ho^2r^2N}{2\EG(o\mu N)^2}
=\frac{2hr^2}{\EG\mu^2}\frac{1}{N},
\end{eqnarray}
where we used (\ref{LROap}).%
\end{proof}
%As an immediate consequence of (\ref{1=PsiA}), we see that for any
%operator $B$ and any normalized state $\Phi'$,
%\begin{equation}
% \abs{\amp{1}{B}{1}(\Psi,B\,\Psi)}\le\frac{c\norm{B}}{\sqrt{N}},
% \label{1=PsiB}
%\end{equation}
%and
%\begin{equation}
% \abs{(\Phi^{(1)},B\,\Phi')(\Psi,B\,\Phi')}\le\frac{c\norm{B}}{\sqrt{N}},
% \label{1=PsiC}
%\end{equation}
%with an $N$independent constant $c$.
We now turn to the estimate of the lefthand side of (\ref{nofluc}).
Note that we can assume
\begin{equation}
\amp{0}{\AO}{0}=0,
\label{0A0=0}
\end{equation}
since otherwise we can redefine $A\amp{0}{A}{0}$ as a new $A$.
Let
\begin{equation}
\Xi':=\frac{1}{\sqrt{2}}(\Phi^{(0)}+\Phi^{(1)}),
\label{Xi'}
\end{equation}
which is essentially the same as $\Xi$ according to the definition
(\ref{XiAp}) and the relation (\ref{1=PsiA}).
In particular, we have
\begin{equation}
\absbar\cbk{(\Xi,\AOS\,\Xi)(\Xi,\AO\,\Xi)^2}
\cbk{(\Xi',\AOS\,\Xi')(\Xi',\AO\,\Xi')^2}\absbar
\le\frac{a_1\norm{A}^2}{\sqrt{N}}\Omega^2.
\label{XivsXi'}
\end{equation}
Throughout the present proof, $a_i$ denote constants which depend
only on $h$, $r$, $\mu$, and $\EG$.
By using the definition (\ref{Xi'}) and the requirement
(\ref{0A0=0}), we observe that
\begin{eqnarray}
&&(\Xi',\AOS\,\Xi')(\Xi',\AO\,\Xi')^2
\ret
&=&\frac{1}{2}\cbk{\amp{0}{\AOS}{0}+\amp{0}{\AOS}{1}
+\amp{1}{\AOS}{0}+\amp{1}{\AOS}{1}}
\ret
&&\frac{1}{4}\cbk{\amp{0}{\AO}{1}+\amp{1}{\AO}{0}
+\amp{1}{\AO}{1}}^2
\ret
&=&\frac{1}{2}\cbk{\amp{0}{\AOS}{0}
\amp{0}{\AO}{1}\amp{1}{\AO}{0}}
\ret
&+&\frac{1}{2}\cbk{\amp{0}{\AOS}{1}
\amp{0}{\AO}{1}\amp{1}{\AO}{1}}
\ret
&+&\frac{1}{2}\cbk{\amp{1}{\AOS}{0}
\amp{1}{\AO}{1}\amp{1}{\AO}{0}}
\ret
&+&R.
\label{=R}
\end{eqnarray}
The remaining term $R$ can be further rewritten as
\begin{eqnarray}
R&=&\frac{1}{2}\amp{1}{\AOS}{1}
\frac{1}{4}\cbk{\amp{0}{\AO}{1}^2+\amp{1}{\AO}{0}^2
+\amp{1}{\AO}{1}^2}
\ret
&=&\frac{1}{2}\cbk{\amp{1}{\AOS}{1}
\sum_{i=0,1}\amp{1}{\AO}{i}\amp{i}{\AO}{1}}
\ret
&+&\frac{1}{4}\cbk{\amp{1}{\AO}{0}\amp{0}{\AO}{1}
\amp{1}{\AO}{0}^2}
\ret
&+&\frac{1}{4}\cbk{\amp{1}{\AO}{0}\amp{0}{\AO}{1}
\amp{0}{\AO}{1}^2}
\ret
&+&\frac{1}{4}\amp{1}{\AO}{1}^2.
\label{R=}
\end{eqnarray}
By using the ``completeness'' relation
(\ref{EGbound}) and (\ref{0A0=0}) to bound the righthand sides of
(\ref{=R}) and (\ref{R=}), we have
\begin{eqnarray}
&&\abs{(\Xi',\AOS\,\Xi')(\Xi',\AO\,\Xi')^2}
\ret
&\le&\frac{1}{2}\abs{\amp{1}{\AO}{0}\amp{0}{\AO}{1}
\amp{1}{\AO}{0}^2}
\ret
&+&\frac{1}{4}\amp{1}{\AO}{1}^2
\ret
&+&\frac{1}{\EG}\norm{[\AO,[H,\AO]]}.
\label{fluc1}
\end{eqnarray}
We shall bound each term in the righthand side of (\ref{fluc1}).
To bound the first term, we use (\ref{1=PsiA}) to get
\begin{eqnarray}
&&\abs{\amp{1}{\AO}{0}\amp{0}{\AO}{1}\amp{1}{\AO}{0}^2}
\ret
&\le&\norm{\AO}\abs{\amp{0}{\AO}{1}\amp{1}{\AO}{0}}
\ret
&\le&\norm{\AO}\cbk{\abs{
\frac{\amp{0}{(AOOA)}{0}}{\amp{0}{O^2}{0}^{1/2}}}
+\frac{a_2\norm{\AO}}{\sqrt{N}}}
\ret
&\le&\frac{\norm{\AO}\norm{[\AO,O]}}{\amp{0}{O}{0}^{1/2}}
+\frac{a_2\norm{\AO}^2}{\sqrt{N}}
\ret
&\le&\frac{2\norm{A}^2r'\Omega^2}{\mu N}
+\frac{a_2\norm{A}^2\Omega^2}{\sqrt{N}},
\label{first}
\end{eqnarray}
where we have used the lower bound (\ref{LROap}) and the bound
$\norm{[\AO,O]}\le2\norm{A}or'\Omega$.
To bound the second term, we first use (\ref{1=PsiA}) and
(\ref{LROap}) to get
\begin{eqnarray}
\amp{1}{\AO}{1}&\le&\frac{\amp{0}{O\AO O}{0}}{\amp{0}{O^2}{0}}
+\frac{a_3\norm{\AO}}{\sqrt{N}}
\ret
&\le&\frac{\amp{0}{O\AO O}{0}}{(\mu o N)^2}
+\frac{a_3\norm{A}\Omega}{\sqrt{N}}.
\label{1A1}
\end{eqnarray}
To bound the righthand side of (\ref{1A1}), we use the ``completeness''
relation (\ref{EGbound}) and (\ref{0A0=0}) to get
\begin{eqnarray}
&&\abs{\amp{0}{O\AO O}{0}}
\le\abs{\amp{0}{O^2\AO}{0}}+\abs{\amp{0}{O[\AO,O]}{0}}
\ret
&&\le\abs{\amp{0}{O^2}{1}\amp{1}{\AO}{0}}
+\frac{\sqrt{\norm{[O^2,[H,O^2]]}\norm{[\AO,[H,\AO]]}}}{2\EG}
+\norm{O[\AO,O]}
\ret
&&\le\abs{\amp{0}{O^2}{1}\norm{\AO}}
+\frac{\sqrt{16o^4h^2rN^3\times4\norm{A}^2hrr'^2\Omega}}{2\EG}
+\norm{A}o^2r'\OmegaN.
\label{OAO}
\end{eqnarray}
We further use (\ref{1=PsiA}) to see
\begin{eqnarray}
\abs{\amp{0}{O^2}{1}}&\le&
\frac{\abs{\amp{0}{O^2O}{0}}}{\amp{0}{O^2}{0}^{1/2}}
+\frac{a_2\norm{O^2}}{\sqrt{N}}
\ret
&\le&o^2a_2N^{3/2},
\label{0OO1}
\end{eqnarray}
where we used (\ref{Ohat3}).
By substituting (\ref{OAO}) and (\ref{0OO1}) into (\ref{1A1}), we get
\begin{equation}
\amp{1}{\AO}{1}^2\le c\frac{\Omega^2}{N},
\label{1A1final}
\end{equation}
where $c$ is an $N$independent constant.
In order to control the third term in the righthand side of (\ref{fluc1}),
we note that
$\norm{[\AO,[H,\AO]]}\le4\norm{A}^2hrr'^2\Omega$.
By putting (\ref{XivsXi'}), (\ref{fluc1}), (\ref{first}),
and (\ref{1A1final}) together, we finally see that
\begin{equation}
\frac{1}{\Omega^2}\abs{(\Xi,\AOS\,\Xi)(\Xi,\AO\,\Xi)^2}
\le \frac{4\norm{A}^2hrr'^2}{\EG}\frac{1}{\Omega}+O(N^{1/2}).
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Lower bound for fluctuation of bulk quantities}
\label{APfluc}
In the present Appendix, we prove simple lemmas which characterize
the behavior of
fluctuation of bulk quantities in a translation invariant state.
The lemma was used in Sections~1.2 and 2.5 to demonstrate that
the infinite volume ground state obtained as a
limit of finite volume ground states is not pure.
Let $\Lambda$ be a $d$dimensional hypercubic
lattice with $N$ sites and with periodic boundary conditions.
We consider a quantum manybody system on $\Lambda$ with
the Hilbert space (\ref{Hilb}).
We do not make any specific assumption about the system.
We denote by $\tau_x$ the translation which acts on the operators
and which respects the periodic boundary conditions.
Let $B$ be an arbitrary local operator.
For a subset $\Omega\subset\Lambda$, we set
\begin{equation}
B_\Omega:=\sum_{x\in\Omega}\tau_x(B).
\end{equation}
\begin{lemma}
Let $\PL$ be an arbitrary state which defines a translation invariant
expectation values, i.e., $\bkt{\tau_x(A)}=\bkt{A}$ for any
local operator $A$.
Then for any local operator $B$, we have
\begin{equation}
\frac{1}{\Omega^2}\bkt{\BO^*\BO}
\ge\frac{1}{N^2}\bkt{\BL^*\BL}.
\label{flucBound}
\end{equation}
\label{flucLemma1}
\end{lemma}
Although we only apply the inequality to ground states in the
present paper, we note that it has a trivial extension to
finite temperature Gibbs states as follows.
We note that the following result has been (implicitly) assumed
in our previous publication \cite{KomaTasaki}, when we mentioned
that the naive infinite volume limit of
the Gibbs states without
symmetry breaking is not pure.
\begin{lemma}
Let $\ham$ be a translation invariant Hamiltonian.
Then for any local operator $B$, we have
\begin{equation}
\frac{1}{\Omega^2}Z(\beta)^{1}{\rm Tr}
\sbk{\BO^*\BO e^{\beta\ham}}
\ge
\frac{1}{N^2}Z(\beta)^{1}{\rm Tr}
\sbk{\BL^*\BL e^{\beta\ham}},
\end{equation}
where the partition function is
$Z(\beta)={\rm Tr}\sbk{\exp[\beta\ham]}$.
\label{flucLemma2}
\end{lemma}
We now prove Lemma~\ref{flucLemma1}.
Let $\calP_B$ be the projection operator onto the state
$\BL\PL/\norm{\BL\PL}$.
If the state is vanishing, we set $\calP_B=0$.
Since $1\calP_B$ is nonnegative, we see that
\begin{eqnarray}
\bkt{\BO^*\BO}
&\ge&\bkt{\BO^*\calP_B\BO}
=\frac{\bkt{\BO^*\BL}\bkt{\BL^*\BO}}{\bkt{\BL^*\BL}}
\ret
&=&\frac{\Omega^2}{N^2}\bkt{\BL^*\BL},
\end{eqnarray}
where we used the translation invariance and the periodic
boundary conditions to get the final line.
If $\calP_B=0$, the inequality is trivial since the
final expression is vanishing.
This proves the lemma.
In order to prove Lemma~\ref{flucLemma2}, we let $\{\Phi^{(n)}\}$
be a complete basis where each basis state $\Phi^{(n)}$
is an eigenstate of $\ham$ with the eigenvalue $E_n$, and
also defines translation invariant expectation values.
By using the bound (\ref{flucBound}), we get
\begin{eqnarray}
Z(\beta)^{1}{\rm Tr}\cbk{\BO^*\BO e^{\beta\ham}}
&=& Z(\beta)^{1}\sum_n\amp{n}{\BO^*\BO}{n}e^{\beta E_n}
\ret
&\ge& \frac{\Omega^2}{N^2}Z(\beta)^{1}\sum_n
\amp{n}{\BL^*\BL}{n}
e^{\beta E_n}
\ret
&=& \frac{\Omega^2}{N^2}Z(\beta)^{1}
{\rm Tr}\sbk{\BL^*\BL e^{\beta\ham}}.
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\par\bigskip\bigskip\bigskip
\noindent{\bf Acknowledgments}
\par\noindent
We wish to thank
Ian Affleck,
Tom Kennedy,
Kenn Kubo,
Elliott Lieb,
Seiji Miyashita,
Tsutomu Momoi,
Bruno Nachtergaele,
Hidetoshi Nishimori,
and
Ken'ichi Takano
for useful discussions on various related topics.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
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